In this talk I will present work motivated by the derivation of the \(\mathbb P_{max}\) axiom \((*)\) from Martin's Maximum\(^{++}\).
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: David Asperó (University of East Anglia, Inglaterra) - titulo: Group operations and universal minimal flows abstract:Every topological group admits a unique, up to isomorphism, universal minimal that maps onto every minimal (with respect to inclusion) flow. We study interactions between group operations and corresponding universal minimal flows.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Dana Bartosova (University of Florida, Estados Unidos) - titulo: Preservation of some covering properties by elementary submodels abstract: |Given a topological space \((X, \tau)\) and an elementary submodel \(M\), we can define the topological space \(X_M = (X\cap M, \tau _M)\), where \(\tau _M\) is the topology on \(X \cap M\) generated by \(\{ V\cap M : V \in \tau \cap M \}\). It is natural to ask which topological properties are preserved by this new operation. For instance, if \(X\) is \(T_2\), then \(X_M\) is also \(T_2\). On the other hand, \(X_M\) compact implies \(X\) compact. A systematic study of it was initiated by L. Junqueira and F. Tall in 1998.
In the paper ``More reflection in topology'', published in Fudamenta Mathematicae in 2003, F. Tall and L. Junqueira, studied the reflection of compactness and, more specifically, when can we have, for \(X\) compact, \(X_M\) compact non trivially, {\it i.e.}, with \(X \neq X_M\). It is natural to try to extend this study for other covering properties.
We will present some results concerning the preservation of Lindelöfness. We will also discuss the perservation of some of its strengthenings, like the Menger and Rothberger properties.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Lucia Junqueira (Universidade de São Paulo, Brasil) joint with Robson A. Figueiredo and Rodrigo R. Carvalho - titulo: The Katetov order on MAD families abstract:The Katetov order is a powerful tool for studying ideals on countable sets. It is specially interesting when restricted to the class of ideals generated by MAD families. One of the reasons we are interested in it is because it allows us to study the destructibility of MAD families under certain forcing extensions. In this talk, I will survey the main known results regarding the Katetov order on MAD families and state some open problems.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Osvaldo Guzmán (Universidad Nacional Autónoma de México, México) - titulo: Hereditary interval algebras and cardinal characteristics of the continuum abstract: |An interval algebra is a Boolean algebra which is isomorphic to the algebra of finite unions of half-open intervals, of a linearly ordered set. An interval algebra is hereditary if every subalgebra is an interval algebra. We answer a question of M. Bekkali and S. Todorcevic, by showing that it is consistent that every \(\sigma\)-centered interval algebra of size \(\mathfrak{b}\) is hereditary. We also show that there is, in ZFC, an hereditary interval algebra of cardinality \(\aleph_1\).
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Carlos Martinez-Ranero (Universidad de Concepción, Chile) - titulo: Groups definable in partial differential fields with an automorphism abstract: |Model theory is a branch of mathematical logic with strong interactions with other branches of mathematics, including algebra, geometry and number theory.
In this talk we are interested in differential and difference fields from the model-theoretic point of view.
A differential field is a field with a set of commuting derivations and a difference-differential field is a differential field equipped with an automorphism which commutes with the derivations.
The model theoretic study of differential fields with one derivation, in characteristic \(0\) started with the work of Abraham Robinson and of Lenore Blum. For several commuting derivations, Tracey McGrail showed that the theory of differential fields of characteristic zero with \(m\) commuting derivations has a model companion called \(DCF\). This theory is complete, \(\omega\)-stable and eliminates quantifiers and imaginaries.
In the case of difference-differential fields, Ronald Bustamante Medina (for the case of one derivation) and Omar León Sánchez (for the general case) showed that the theory of difference-differential fields with \(m\) derivations admits a model companion called \(DCF_mA\). This theory is model-complete, supersimple and eliminates imaginaries.
Cassidy studied definable groups in models of \(DCF\), in particular she studied Zariski dense definable subgroups of simple algebraic groups and showed that they are isomorphic to the rational points of an algebraic group over some definable field.
In this talk we study groups definable in models of \(DCF_mA\), and show an analogue of Phyllis Cassidy's result.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Samaria Montenegro (Universidad de Costa Rica, Costa Rica), joint work with Ronald Bustamente Medina and Zoé Chatzidakis - titulo: Some lessons after the formalization of the ctm approach to forcing abstract: |In this talk we'll discuss some highlights of our computer-verified proof of the construction, given a countable transitive set model \(M\) of \(\mathit{ZFC}\), of a generic extension \(M[G]\) satisfying \(\mathit{ZFC}+\neg\mathit{CH}\). In particular, we isolated a set \(\Delta\) of \(\sim\)220 instances of the axiom schemes of Separation and Replacement and a function \(F\) such that such that for any finite fragment \(\Phi\subseteq\mathit{ZFC}\), \(F(\Phi)\subseteq\mathit{ZFC}\) is also finite and if \(M\models F(\Phi) + \Delta\) then \(M[G]\models \Phi + \neg \mathit{CH}\). We also obtained the formulas yielded by the Forcing Definability Theorem explicitly.
To achieve this, we worked in the proof assistant Isabelle, basing our development on the theory Isabelle/ZF by L. Paulson and others.
The vantage point of the talk will be that of a mathematician but elements from the computer science perspective will be present. Perhaps some myths regarding what can effectively be done using proof assistants/checkers will be dispelled.
We'll also compare our formalization with the recent one by Jesse M. Han and Floris van Doorn in the proof assistant Lean.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Pedro Sánchez Terraf (Universidad Nacional de Córdoba, Argentina) joint with Emmanuel Gunther, Miguel Pagano, and Matías Steinberg - titulo: On non-classical models of ZFC abstract: |In this talk we present recent developments in the study of non-classical models of ZFC.
We will show that there are algebras that are neither Boolean, nor Heyting, but that still give rise to models of ZFC. This result is obtained by using an algebra-valued construction similar to that of the Boolean-valued models. Specifically we will show the following theorem.
There is an algebra \(\mathbb{A}\), whose underlying logic is neither classical, nor intuitionistic such that \(\mathbf{V}^{\mathbb{A}} \vDash\) ZFC. Moreover, there are formulas in the pure language of set theory such that \(\mathbf{V}^{\mathbb{A}} \vDash \varphi \land \neg \varphi\).
The above result is obtained by a suitable modification of the interpretation of equality and belongingness, which are classical equivalent to the standard ones, used in Boolean-valued constructions.
Towards the end of the talk we will present an application of these constructions, showing the independence of CH from non-classical set theories, together with a general preservation theorem of independence from the classical to the non-classical case.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Giorgio Venturi (Universidade Estadual de Campinas, Brasil), joint work with Sourav Tarafder and Santiago Jockwich - sesion: Combinatoria algebraica de funciones simétricas, cuasisimétricas y sus generalizaciones zoom: https://us02web.zoom.us/j/86548084835 organizadores: - nombre: Rafael S. González D'León mail: rafael.gonzalezl@usa.edu.co - nombre: Yannic Vargas Lozada mail: yannicmath@gmail.com charlas: - titulo: Hopf monoids of type B, their antipode and examples abstract:Combinatorial species provide a unified framework to study families of combinatorial objects. If the family of combinatorial objects has natural operations to merge and break structures, the corresponding species becomes a Hopf monoid. In this talk, we present a novel definition of type B Hopf monoids. In the same spirit as the work of Bergeron and Choquette, we consider structures over finite sets with a fixed-point free involution. However, instead of defining a monoidal structure on the category of type B species, our construction involves an action of the monoidal category of (standard) species on the category of Type B species. We present the basic definitions and some examples of Type B Hopf monoids constructed from (type B) set compositions, (symplectic) matroids, and (type B) generalized permutahedra. This is work in progress with M. Aguiar.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: José Bastidas (LACIM/Université du Québec à Montréal, Canadá) - titulo: Chromatic symmetric functions for Dyck paths and \(q\)-rook theory abstract: |Given a graph and a set of colors, a coloring is a function that associates each vertex in the graph with a color. In 1995, Stanley generalized this definition to symmetric functions by looking at the number of times each color is used and extending the set of colors to \(\mathbb{Z}^+\). In 2012, Shareshian and Wachs introduced a refinement of the chromatic functions for ordered graphs as \(q\)-analogues.
In the particular case of Dyck paths, Stanley and Stembridge described the connection between chromatic symmetric functions of abelian Dyck paths and square hit numbers, and Guay-Paquet described their relation to rectangular hit numbers. Recently, Abreu-Nigro generalized the former connection for the Shareshian-Wachs \(q\)-analogue, and in unpublished work, Guay-Paquet generalized the latter.
In this talk, I want to give an overview of the framework and present another proof of Guay-Paquet's identity using \(q\)-rook theory. Along the way, we will also discuss \(q\)-hit numbers, two variants of their statistic, and some deletion-contraction relations. This is recent work with Alejandro H. Morales and Greta Panova.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Laura Colmenarejo (North Carolina State University, Estados Unidos) - titulo: Peak algebra for combinatorial Hopf algebras abstract:The peak algebra is originally introduced by Stembridge using enriched \(P\)-partitions. Using the character theory by Aguiar-Bergeron-Sottile, the peak algebra is also the image of \(\Theta\), the universal morphism between certain combinatorial Hopf algebras. We introduce a shuffle basis of quasi-symmetric functions that has a shuffle-like Hopf structure and is also the eigenfunctions of \(\Theta\). Using this new basis, we extend the notion of peak algebras and theta maps to shuffle, tensor and symmetric algebras. As examples, we study the peak algebras of symmetric functions in non-commuting variables and the graded associated Hopf algebra on permutations.
start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Shu Xiao Li (Dalian University of Technology, China) - titulo: A multiset partition algebra abstract:Classical Howe duality provides a representation theoretic framework for classical invariant theory. In the classical Howe duality the general linear group \(GL_n\) is dual to \(GL_k\) when acting on the polynomial ring on variables \(x_{ij}\) where \(1\leq i\leq n\) and \(1\leq j \leq k\). In this talk we restrict the action of \(GL_n\) to the group of permutation matrices and show that the Howe dual is an algebra whose basis is indexed by multiset partition algebra.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Rosa C. Orellana (Dartmouth College, Estados Unidos) - titulo: Valuations and the Hopf Monoid of Generalized Permutahedra abstract:Many combinatorial objects, such as matroids, graphs, and posets, can be realized as generalized permutahedra - a beautiful family of polytopes. This realization respects the natural multiplication of these objects as well as natural "breaking" operations. Surprisingly many of the important invariants of these objects, when viewed as functions on polytopes, satisfy an inclusion-exclusion formula with respect to subdivisions. Functions that satisfy this formula are known as valuations. In this talk, I will discuss recent work with Federico Ardila that completely describes the relationship between the algebraic structure on generalized permutahedra and valuations. Our main contribution is a new easy-to-apply method that converts simple valuations into more complicated ones. We also describe an universality property of the Hopf monoid of indicator functions of generalized permutahedra that extends the relationship between Combinatorial Hopf algebras and QSYM.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Mario Sanchez (UC Berkeley, Estados Unidos) - titulo: Computation of Kronecker coefficients abstract: |A Kronecker coefficient is a non-negative integer that depends on three partitions of a natural number n. It is the multiplicity of an irreducible representation of the symmetric group of degree n in the tensor (or Kronecker) product of two other irreducible representations of the same group.
The study of ways of computing Kronecker coefficients is an important topic on algebraic combinatorics. It was initiated by Francis Murnaghan more than eighty years ago. Several tools have been used to try to understand them, notably from representation theory, symmetric functions theory and Borel-Weil theory. These numbers generalize the well-known Littlewood-Richardson coefficients, but are still very far to be fully grasped.
It is known that each Kronecker coefficient can be described as an alternating sum of numbers of integer points in convex polytopes.
In this talk we present a new family of polytopes that permit very fast computations on Kronecker coefficients associated to partitions with few parts. This family provides, in particular, insight into some properties of Kronecker coefficients as well as into Murnaghan stability.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Ernesto Vallejo (Universidad Nacional Autónoma de México, México) - titulo: The Castelnuovo-Mumford Regularity of Matrix Schubert Varieties abstract:The Castelnuovo-Mumford regularity of a graded module provides a measure of how complicated its minimal free resolution is. In work with Rajchogt, Ren, Robichaux, and St. Dizier, we noted that the CM-regularity of matrix Schubert varieties can be easily obtained by knowing the degree of the corresponding Grothendieck polynomial. Furthermore, we gave explicit, combinatorial formulas for these degrees for symmetric Grothendieck polynomials. In this talk, I will present a general degree formula for Grothendieck polynomials. This is joint work with Oliver Pechenik and David Speyer.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Anna E. Weigand (Massachusetts Institute of Technology, Estados Unidos) - titulo: 'Chromatic symmetric homology for graphs: some new developments' abstract: |In his study of the four colour problem, Birkhoff showed that the number of ways to colour a graph with k colours is a polynomial chi(k), which he called the chromatic polynomial. Later, Stanley defined the chromatic symmetric function X(x_1, x_2, ... ), which is a multivariable lift of the chromatic polynomial so that when the first k variables are set to 1, it recovers chi(k). This can be further lifted to a homological setting; we can construct a chain complex of graded S_n-modules whose homology has a bigraded Frobenius characteristic that recovers X upon setting q=t=1.
In this talk, we will explain the construction of the homology, discuss some new results regarding the strength of the homology as a graph invariant, and state some surprising conjectures regarding integral symmetric homology for graphs. This is based on joint work with Chandler, Sazdanovic, and Stella.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Martha Yip (University of Kentucky, Estados Unidos) - sesion: Lógica y Computación zoom: https://us02web.zoom.us/j/82707046662 organizadores: - nombre: Alexandre Miquel mail: amiquel@fing.edu.uy - nombre: Martin Hyland mail: M.Hyland@dpmms.cam.ac.uk charlas: - titulo: Random! abstract:Everyone has an intuitive idea about what is randomness, often associated with ``gambling'' or ``luck''. Is there a mathematical definition of randomness? Are there degrees of randomness? Can we give examples of randomness? Can a computer produce a sequence that is truly random? What is the relation between randomness and logic? In this talk I will talk about these questions and their answers.
start: 2021-09-14T16:45-0300 end: 2021-09-14T17:30-0300 speaker: Verónica Becher (Universidad de Buenos Aires, Argentina) - titulo: Relating logical approaches to concurrent computation abstract: |This talk will present ongoing work towards the description and study of concurrent interaction in proof theory.
Type systems that are designed to ensure behavioural properties of concurrent processes (input/output regimes, lock-freeness) generally have unclear logical meanings. Conversely, proofs-as-programs correspondences for processes (e.g. with session types) tend to impose very functional behaviour and little actual concurrency. Besides, relationships between type systems and denotational models of concurrency are rarely established.
A possible reason for this state of things is the ambiguous status of non-determinism in logic and the importance of scheduling concerns in models of concurrency, to which traditional proof theory is not accustomed. Unifying logical approaches in a consistent framework requires to put a focus on these issues, and this talk will propose, building on recent developments in proof theory, in the veins of linear logic and classical realizability.
start: 2021-09-13T15:00-0300 end: 2021-09-13T15:45-0300 speaker: Emmanuel Beffara (Université Grenoble Alpes, Francia) - titulo: A framework to express the axioms of mathematics abstract: 'The development of computer-checked formal proofs is a major step forward in the endless quest for mathematical rigor. But it also has a negative aspect: the multiplicity of systems brought a multiplicity of theories in which these formal proofs are expressed. We propose to define these theories in a common logical framework, called Dedukti. Some axioms are common to the various theories and some others are specific, just like some axioms are common to all geometries and some others are specific. This logical framework extends predicate logic in several ways and we shall discuss why predicate logic must be extended to enable the expression of these theories.
' start: 2021-09-14T15:45-0300 end: 2021-09-14T16:30-0300 speaker: Gilles Dowek (Institut de la Recherche en Informatique et Automatique, Francia) - titulo: Generalized Algebraic Theories and Categories with Families abstract:
We give a new syntax independent definition of the notion of a finitely presented generalized algebraic theory as an initial object in a category of categories with families (cwfs) with extra structure. To this end we define inductively how to build a valid signature \(\Sigma\) for a generalized algebraic theory and the associated category \(\textrm{CwF}_{\Sigma}\) of cwfs with a \(\Sigma\)-structure and cwf-morphisms that preserve \(\Sigma\)-structure on the nose. Our definition refers to the purely semantic notions of uniform family of contexts, types, and terms. Furthermore, we show how to syntactically construct initial cwfs with \(\Sigma\)-structures. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of dependent type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual categories with families.
start: 2021-09-13T15:45-0300 end: 2021-09-13T16:30-0300 speaker: Peter Dybjer (Chalmers University of Technology, Suecia), joint with Marc Bezem, Thierry Coquand, and Martin Escardo - titulo: On the instability of the consistency operator abstract:We examine recursive monotonic functions on the Lindenbaum algebra of EA. We prove that no such function sends every consistent \(\varphi\), to a sentence with deductive strength strictly between \(\varphi\) and \(\textit{Con}(\varphi)\). We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function \(f\), if there is an iterate of \(\textit{Con}\) that bounds \(f\) everywhere, then \(f\) must be somewhere equal to an iterate of \(\textit{Con}\).
start: 2021-09-13T17:30-0300 end: 2021-09-13T18:15-0300 speaker: Antonio Montalbán (Berkeley University of California, Estados Unidos), joint work with James Walsh - titulo: Readers by name, presheaves by value abstract:Presheaves are an ubiquitary model construction used everywhere in logic, particularly in topos theory. It is therefore tempting to port them to the similar but slightly different context of type theory. Unfortunately, it turns out that there are subtle issues with the built-in computation rules of the latter, which we will expose. As an alternative, we will describe a new structure that is much better behaved in an intensional setting, but categorically equivalent to presheaves in an extensional one. Such a structure is motivated by considerations stemming from the study of generic side-effects in programming language theory, shedding a new light on the fundamental nature of such a well-known object.
start: 2021-09-14T15:00-0300 end: 2021-09-14T15:45-0300 speaker: Pierre-Marie Pédrot (Institut de la Recherche en Informatique et Automatique, Francia) - titulo: Reversible computation and quantum control abstract:One perspective on quantum algorithms is that they are classical algorithms having access to a special kind of memory with exotic properties. This perspective suggests that, even in the case of quantum algorithms, the control flow notions of sequencing, conditionals, loops, and recursion are entirely classical. There is however, another notion of control flow, that is itself quantum. This purely quantum control flow is however not well-understood. In this talk, I will discuss how to retrieve some understanding of it with a detour through reversible computation. This will allow us to draw links with the logic \(\mu\)MALL, pointing towards a Curry-Howard isomorphism.
start: 2021-09-13T16:45-0300 end: 2021-09-13T17:30-0300 speaker: Benoît Valiron (CentraleSupélec, Francia) - sesion: Algebraic and categorical structures in geometry and topology zoom: https://us02web.zoom.us/j/88099466470 organizadores: - nombre: Manuel Rivera mail: manuelor@gmail.com - nombre: Camilo Arias Abad mail: carias0@unal.edu.co charlas: - titulo: Discretization of euclidean space ,vector calculus and 3D incompressible fluids abstract:Overlapping cubical decompositions admit hodge star dualities and analogues of Grassman algebra for differential forms and multivector fields with their exterior d and the divergence operator differentials del. For these operators there are hierarchies of deformation corrections to the first order derivation structure of exterior d and the second order derivation structure of the divergence operator. These suggest computer codes with surprising properties. There is a joint paper in the Atiyah Memorial Volume 2021 with Ruth Lawrence and Nissim Ranade describing this discretization with explicit calculations of the deformations.The computer studies are in progress.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Dennis Sullivan (Stony Brook University and City University of New York Graduate Center, Estados Unidos) - titulo: Objective combinatorial bialgebras through decomposition spaces abstract:Decomposition spaces (also known as 2-Segal spaces) and the machinery of homotopy linear algebra allow us to realise objective combinatorial bialgebra structures, rendering classical combinatorial bialgebras once cardinalites are taken. In many cases such bialgebras are related, through CULF functors given by base change, Galois connection, duality, that descend to their numerical counterparts but provide deeper understanding of phenomena at the objective level. We will present a few important cases of such constructions, in particular relating to the bialgebras of Malvenuto-Reutenauner and of symmetric functions. Joint work in progress with Joachim Kock and Andrew Tonks.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Imma Gálvez-Carrillo (Universitat Politècnica de Catalunya, España) - titulo: Variants of the Waldhausen S-construction abstract:The S-construction, first defined in the setting of cofibration categories by Waldhausen, gives a way to define the algebraic K-theory associated to certain kinds of categorical input. It was proved by Galvez-Carrillo, Kock, and Tonks that the result of applying this construction to an exact category is a decomposition space, also called a 2-Segal space, and Dyckerhoff and Kapranov independently proved the same result for the slightly more general input of proto-exact categories. In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we proved that these results can be maximally generalized to the input of augmented stable double Segal spaces, so that the S-construction defines an equivalence of homotopy theories. In this talk, we'll review the S-construction and the reasoning behind these stages of generalization. Time permitting, we'll discuss attempts to characterize those augmented stable double Segal spaces that correspond to cyclic spaces, which is work in progress with Walker Stern.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Julie Bergner (University of Virginia, Estados Unidos) - titulo: Transfer systems and weak factorization systems abstract:N∞ operads over a group G encode homotopy commutative operations together with a class of equivariant transfer (or norm) maps. Their homotopy theory is given by transfer systems, which are certain discrete objects that have a rich combinatorial structure defined in terms of the subgroup lattice of G. In this talk, we will show that when G is finite Abelian, transfer systems are in bijection with weak factorization systems on the poset category of subgroups of G. This leads to an involution on the lattice of transfer systems, generalizing the work of Balchin-Bearup-Pech-Roitzheim for cyclic groups of squarefree order. This is joint work with Evan Franchere, Usman Hafeez, Peter Marcus, Kyle Ormsby, Weihang Qin, and Riley Waugh.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Angélica Osorno (Reed College, Estados Unidos) - titulo: Classifying stacky vector bundles abstract:Lie groupoids up to Morita equivalences serve as models for differentiable stacks, categorified spaces that generalize manifolds and orbifolds, and which are useful when dealing with singular quotients. Vector bundles over Lie groupoids have the tangent and cotangent constructions as prominent examples, and they admit a nice interpretation in terms of representations up to homotopy. In this talk, based on joint works with C. Ortiz, D. Stefani, and J. Desimoni, I will first discuss the Morita invariance of vector bundles, then describe the general linear 2-groupoid, and finally present a classification of stacky vector bundles by the resulting 2-Grassmannian.
start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Matías del Hoyo (Universidade Federal Fluminense, Brasil) - titulo: Cut cotorsion pairs abstract: |In this talk, I shall present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations of objects restricted to certain subcategories. Several applications will be given in the settings of relative Gorenstein homological algebra and chain complexes, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.
This is a joint work with Mindy Huerta and Octavio Mendoza (Instituto de Matemáticas - Universidad Nacional Autónoma de México).
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Marco Perez (Universidad de la República, Uruguay) - titulo: Hopf and Bialgebras in Algebra, Topology and Physics abstract: |Together with Imma Galvez-Carillo and Andy Tonks, we constructed Bi and Hopf-algebras from a three-fold hierarchy, simplicial objects, Co-operads with multiplication and Feynman categories. The common theme is that the co-product is a dualized composition product and the product iis an extra structure, which is free in the most prominent examples. These are the Hopf algebra of Baues in Topology, those of Goncharov and Brown in number theory, and those of Connes and Kreimer in mathematical physics. The natural structures are bi-algebras and the Hopf algebras appear as a connected quotient.
Together with Yang Mo, we introduced the notion of a path-like bi-algebra in order to better understand the relationship between the bialgebras and the Hopf algebras, which allowed us to complete the picture. The usual Quillen condition of being connected is replaced by the so-called Quillen-Takeushi filtration being exhaustive. This allows us to reduce the obstructions to having an antipode to (semi)-grouplike elements. This theory comprises all the examples above as well as those of May as special cases. Finally, quotients now appear naturally as the universal quotient spaces through which characters -with particular properties- factor. The characters take values in Rota-Baxter-algebras which are at the heart of the renormalization theory. This brings the theory full circle.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Ralph Kaufmann (Purdue University, Estados Unidos) - titulo: Remarks concerning moduli spaces in quantum toric geometry abstract: | Quantum toric geometry generalizes toric geometry just as the quantum torus generalizes the usual torus. In this talk I will explain how this theory comes about. Also, this generalization of toric geometry is amenable to the definition of new moduli spaces (of quantum toric manifolds) and opens up many interesting questions regarding the geometry and topology of these moduli spaces. This is joint work with Katzarkov, Meersseman and Verjovsky. start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Ernesto Lupercio (Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, México) - sesion: 'Problemas inversos: desde la teoría a las aplicaciones' zoom: https://us02web.zoom.us/j/87211705520 organizadores: - nombre: Adriano Dr Cezaro mail: adrianocezaro@furg.br - nombre: Alberto Mercado mail: alberto.mercado@usm.cl - nombre: Juan Pablo Agnelli mail: agnelli@famaf.unc.edu.ar charlas: - titulo: Synergistic multi-spectral CT reconstruction with directional total variation abstract:This work considers synergistic multi-spectral CT reconstruction where information from all available energy channels is combined to improve the reconstruction of each individual channel. We propose to fuse these available data (represented by a single sinogram) to obtain a polyenergetic image that keeps structural information shared by the energy channels with an increased signal-to-noise ratio. This new image is used as prior information during a channel-by-channel minimization process through the directional total variation. We analyze the use of directional total variation within variational regularization and iterative regularization. Our numerical results on simulated and experimental data show improvements in terms of image quality and in computational speed.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Evelyn Cueva (Escuela Politécnica Nacional, Ecuador) - titulo: Bayesian Approach Helmholtz Inverse Problem abstract:This work present an implementation for solving the Helmholtz inverse problem via Finite Element Methods and Bayesian inference theory. The solution of the inverse problem is a posterior distribution based on a number observation data sets, the forward problem governing the state solution, and a statistical prior distribution which describes the behavior of the uncertainty linked to the parameters.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Lilí Guadarrama (Centro de Investigación en Matemáticas, México) - titulo: Partial data Photoacoustic Tomography in unbounded domains abstract: |Photoacoustic Tomography is a promising hybrid medical imaging modality that is able to generate high-resolution and high-contrast images, by exploiting the coupling of electromagnetic pulses (in the visible region) and ultrasound waves via de photoacoustic effect.
In this talk I will introduce the modality and focus on the ultrasound propagation part which is mathematically modeled as an inverse initial source problem for the wave equation. We will pose this problem in a setting consisting of an unbounded domain with boundary, with observation taking place in a subset of it. We will address the question of uniqueness, stability and reconstruction of the initial source of acoustic waves.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Benjamín Palacios (Pontificia Universidad Católica de Chile, Chile) - titulo: Particle Source Estimation via Analytical Formulations of the Adjoint Flux abstract:Numerous applications of research interest may benefit from the study and development of particle sources estimation methods, such as non-destructive material identification and nuclear safety. In such problems, we are concerned with estimating the spatial distribution and intensity of internal sources of neutral particles. In this context, we use the adjoint to the transport operator to derive a linear model that relates the absorption rate measurements of a series of internal particle detectors with the coefficients of the source expansion on a given basis. We consider both one-dimensional energy-dependent and two-dimensional (\(XY\)-geometry) monoenergetic transport problems. We use the adjoint version of the Analytical Discrete Ordinates (ADO) method to write spatially explicit solutions for the adjoint flux for one-dimensional problems. We consider a nodal version of the adjoint ADO method to writing similar explicit expressions for the \(x\) and \(y\) averaged adjoint fluxes for the two-dimensional case. In both cases, we use explicit expressions to derive closed-form formulas to the absorption rate. Then, we apply the iterated Tikhonov algorithm to estimate the sources from noisy measurements and a Bayesian approach through the Metropolis-Hastings algorithm. We successfully estimated one-dimensional polynomial and piecewise constant localized energy dependent sources and two-dimensional piecewise constant localized sources from noise measurements. In either case, the use of closed-form expressions allowed a relevant reduction in computational time.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Liliane Basso Barichello (Universidade Federal do Rio Grande do Sul, Brasil) - titulo: Inverse scattering for the Jacobi system using input data containing transmission eigenvalues abstract: |The Jacobi system with the Dirichlet boundary condition is considered on a half-line lattice with real-valued coefficients including weight factors. The inverse problem of recovery of the coefficients is analyzed by using certain input data sets containing the transmission eigenvalues.
This is joint work with T. Aktosun of University of Texas at Arlington and V. G. Papanicolaou of National Technical University of Athens.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Abdon Choque Rivero (Universidad Michoacana de San Nicolás de Hidalgo, México) - titulo: Identificación del coeficiente de velocidad local en un modelo bi-direccional para la propagación de ondas en la superficie de un canal con fondo variable abstract: |Un problema inverso es aquel donde se requiere determinar el valor de algunos parámetros de un modelo a partir de datos observados. El estudio de este tipo de problemas es de mucha importancia y aparece en diversas ramas de la ciencia y de las Matemáticas. En esta charla consideramos el problema inverso de identificación del coeficiente de velocidad lineal local en un sistema de tipo Boussinesq linealizado de tipo dispersivo deducido por Quintero y Muñoz (2004), a partir de las mediciones \(V_T(x), N_T(x)\) en un tiempo final \(T\), de la velocidad de las partículas en una profundidad fija y la amplitud de la onda, que se propaga en la superficie de un canal raso con fondo variable.
El problema inverso es reformulado como un problema de optimización con restricciones, donde la funcional objetivo es de tipo usado en métodos de mínimos cuadrados junto con una regularización del tipo introducido por Tikhonov (1963). El proceso de minimización del funcional Tikhonov adoptado se realiza utilizando el método iterativo quasi-Newton L-BFGS-B, introducido por Byrd et al. (1995). Se presentan simulaciones numéricas para mostrar la convergencia y robustez del método numérico propuesto, a\'un en el caso en que el coeficiente posee discontinuidades, o existe ruido en la medici\'on de los datos de entrada \(V_T(x), N_T(x)\).
Investigación realizada con el apoyo de la Universidad del Valle mediante proyecto C.I. 71235.
Referencias:
Tikhonov, A.N. (1963): Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl. 4, 1035-1038.
Byrd, R.H., Lu, P., Nocedal, J. (1995): A limited memory algorithm for Bound Constrained Optimization. SIAM J. Sci. Stat. Comp. 16 (5): 1190-1208.
Quintero, J.R. , Muñoz, J.C. (2004): Existence and uniqueness for a Boussinesq system with a disordered forcing. Methods Appl. Anal., 11 (1): 015-032.
An image inpainting problem consists of restoring an image from a (possibly noisy) observation, in which data from one or more regions is missing. Several inpainting models to perform this task have been developed, and although some of them perform reasonably well in certain types of images, quite a few issues are yet to be sorted out. For instance, if the original image is expected to be smooth, inpainting can be performed with reasonably good results by modeling the solution as the result of a diffusion process using the heat equation. For non-smooth images, however, such an approach is far from being satisfactory. On the other hand, Total Variation (TV) inpainting models based on high order PDE diffusion equations can be used whenever edge restoration is a priority. More recently, the introduction of spatially variant conductivity coefficients on these models, such as in the case of Curvature-Driven Diffusions (CDD), has allowed inpainted images with well defined edges and enhanced object connectivity. The CDD approach, nonetheless, is not suitable wherever the image is smooth, as it tends to produce piecewise constant solutions. Based upon this, we propose using CDD to gather a-priori information used at a second step in a weighted anisotropic mixed Tikhonov plus total- variation model that allows for both edge preservation and object connectivity while precluding the staircasing effect that all pure TV-based methods entail. Comparisons between the results of the implemented models will be illustrated by several computed examples, along with performance measures.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Ruben D. Spies (Instituto de Matemática Aplicada del Litoral, Argentina) - titulo: A Splitting Strategy for the Calibration of Jump-Diffusion Models abstract:We present a splitting strategy to identify simultaneously the local-volatility surface and the jump-size distribution from quoted European prices. The underlying is a jump-diffusion driven asset with time and price dependent volatility. Our approach uses a forward partial-integro-differential equation for the option prices to produce a parameter-to-solution map. The corresponding inverse problem is then solved by Tikhonov-type regularization combined with a splitting strategy.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Vinicus V. Albani (Universidade Federal de Santa Catarina, Brasil) - titulo: Lipschitz Stability for the Backward Heat Equation and its application to Lightsheet Fluorescence Microscopy abstract: |In this talk we will present a result about Lipschitz stability for the reconstruction of a compactly supported initial temperature for the heat equation in \(\mathbb{R}^n\), from measurements along a positive time interval and over an open set containing its support. We apply these results to deduce a Lipschitz stability inequality for the backwards heat equation problem in \(\mathbb{R}\) with measurements on a curve contained in \( \mathbb{R}\times[0,T]\), leading to stability estimates for an inverse problem arising in 2D Fluorescence Microscopy.
This is joint work with Pablo Arratia, Evelyn Cueva, Axel Osses and Benjamin Palacios.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Matias Courdurier (Pontificia Universidad Católica, Chile) - titulo: Anatomical atlas of the upper part of the human head for electroencephalography and bioimpedance applications abstract: |Electrophysiology is the branch of physiology that investigates the electrical properties of biological tissues. Volume conductor problems in cerebral electrophysiology and bioimpedance do not have analytical solutions for nontrivial geometries and require a 3D model of the head and its electrical properties for solving the associated PDEs numerically.
Ideally, the model should be made with patient-specific information. In clinical practice, this is not always the case and an average head model is often used. Also, the electrical properties of the tissues might not be completely known due to natural variability.
A 4D (3D+T) statistical anatomical atlas of the electrical properties of the upper part of the human head for cerebral electrophysiology and bioimpedance applications is presented. The atlas is constructed based on MRI images of human individuals and comprises the electrical properties of the main internal structures and can be adjusted for specific electrical frequencies. The atlas also comprises a time-varying model of arterial brain circulation, based on the solution of the Navier-Stokes equations.
The atlas is an important tool for in silico studies on cerebral circulation and electrophysiology that require statistically consistent data, e.g., machine learning, sensitivity analyses, and as a benchmark to test inverse problem solvers. The atlas can also be used as statistical prior information for inverse problems in electrophysiology.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Fernando Moura da Silva (Universidade Federal do ABC, Brasil y University of Helsinki, Finlandia) - sesion: Estadística Matemática zoom: https://us02web.zoom.us/j/86910588204 organizadores: - nombre: Florencia Leonardi mail: florencia@usp.br - nombre: Pamela Llop mail: lloppamela@gmail.com - nombre: Daniela Rodriguez mail: drodrig@dm.uba.ar charlas: - titulo: Differentially private inference via noisy optimization abstract:We propose a general optimization-based framework for computing differentially private M-estimators and a new method for the construction of differentially private confidence regions. Firstly, we show that robust statistics can be used in conjunction with noisy gradient descent and noisy Newton methods in order to obtain optimal private estimators with global linear or quadratic convergence, respectively. We establish global convergence guarantees, under both local strong convexity and self-concordance, showing that our private estimators converge with high probability to a neighborhood of the non-private M-estimators. The radius of this neighborhood is nearly optimal in the sense it corresponds to the statistical minimax cost of differential privacy up to a logarithmic term. Secondly, we tackle the problem of parametric inference by constructing differentially private estimators of the asymptotic variance of our private M-estimators. This naturally leads to the use of approximate pivotal statistics for the construction of confidence regions and hypothesis testing. We demonstrate the effectiveness of a bias correction that leads to enhanced small-sample empirical performance in simulations.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Marco Avella Medina (Columbia University, Estados Unidos) - titulo: Adaptive regression with Brownian path covariate abstract:In this talk, we will study how to obtain optimal estimators in problems of non-parametric estimation. More specifically, we will present the Goldenshluger-Lepski (2011) method that allows one to obtain estimators that adapt to the smoothness of the function to be estimated. We will show how to extend this statistical procedure in regression with functional data when the regressor variable is a Wiener process \(W\). Using the Wiener-Ito decomposition of m(W), where \(m\) is the regression function, we will define a family of estimators that satisfy an oracle inequality, are proved to be adaptive and converge at polynomial rates over specific classes of functions.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Karine Bertin (Universidad de Valparaíso, Chile) - titulo: 'Adjusting ROC curves for covariates: a robust approach' abstract: |ROC curves are a popular tool to describe the discriminating power of a binary classifier based on a continuous marker as the threshold is varied. They become an interesting strategy to evaluate how well an assignment rule based on a diagnostic test distinguishes one population from the other. Under certain circumstances the marker's discriminatory ability may be affected by certain covariates. In this situation, it seems sensible to include this information in the ROC analysis. This task can be accomplished either by the induced or the direct method.
In this talk we will focus on ROC curves in presence of covariates. We will show the impact of outliers on the conditional ROC curves and we will introduce a robust proposal. We follow a semiparametric approach where we combine robust parametric estimators with weighted empirical distribution estimators based on an adaptive procedure that downweights outliers.
We will discuss some aspects concerning consistency and through a Monte Carlo study we will compare the performance of the proposed estimators with the classical ones both, in clean and contaminated samples.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Ana M. Bianco (Universidad de Buenos Aires, Argentina) - titulo: Least trimmed squares estimators for functional principal component analysis abstract: 'Classical functional principal component analysis can yield erroneous approximations in presence of outliers. To reduce the influence of atypical data we propose two methods based on trimming: a multivariate least trimmed squares (LTS) estimator and its coordinatewise variant. The multivariate LTS minimizes the multivariate scale corresponding to \(h-\)subsets of curves while the coordinatewise version uses univariate LTS scale estimators. Consider a general setup in which observations are realizations of a random element on a separable Hilbert space \(\mathcal{H}\). For a fixed dimension \(q\), we aim to robustly estimate the \(q\) dimensional linear space in \(\mathcal{H}\) that gives the best approximation to the functional data. Our estimators use smoothing to first represent irregularly spaced curves in a high-dimensional space and then calculate the LTS solution on these multivariate data. The solution of the multivariate data is subsequently mapped back onto \(\mathcal{H}\). Poorly fitted observations can therefore be flagged as outliers. Simulations and real data applications show that our estimators yield competitive results when compared to existing methods when a minority of observations is contaminated. When a majority of the curves is contaminated at some positions along its trajectory coordinatewise methods like Coordinatewise LTS are preferred over multivariate LTS and other multivariate methods since they break down in this case.
' start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Holger Cevallos-Valdiviezo (Escuela Superior Politécnica del Litoral, Ecuador y Ghent University, Bélgica) - titulo: A non-asymptotic analysis of certain high-dimensional estimators for the mean abstract:Recent work in Statistics and Computer Science has considered the following problem. Given a distribution \(P\) over \({\bf R}^d\) and a fixed sample size \(n\), how well can one estimate the mean \(\mu = \bf E_{X\sim P} X\) from a sample \(X_1,\dots,X_n\stackrel{i.i.d.}{\sim}P\) while only requiring finite second moments and allowing for sample contamination? It turns out that the best estimators are not related to the sample mean. In this talk we present a new analysis of certain approaches to this problem, and reproduce or improve previous results by several authors.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Roberto Imbuzeiro Oliveira (Instituto de Matemática Pura e Aplicada, Brasil) - titulo: Stick-breaking priors via dependent length variables abstract:In this talk, we present new classes of Bayesian nonparametric prior distributions. By allowing length random variables, in stick-breaking constructions, to be exchangeable or Markovian, appealing models for discrete random probability measures appear. As a result, by tuning the stochastic dependence in such length variables allows to recover extreme families of random probability measures, i.e. Dirichlet and Geometric processes. As a byproduct, the ordering of the weights, in the species sampling representation, can be controlled and thus tuned for efficient MCMC implementations in density estimation or unsupervised classification problems. Various theoretical properties and illustrations will be presented.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Ramsés Mena Chávez (Universidad Nacional Autónoma de México, México) - titulo: 'Modelling in pandemic times: using smart watch data for early detection of COVID-19' abstract: |The COVID-19 pandemic brought many challengers to statisticians and modelers across all quantitative disciplines. From accelerated clinical trials to modelling of epidemiological interventions at a feverish pace, to the study of the impact of the pandemic in mental health outcomes and racial disparities. In this talk, we would like to share our experience using classical statistical modelling and machine learning to use data obtained from wearable devices as digital biomarkers of COVID-19 infection.
Early in the pandemic, health care workers in the Mount Sinai Health System (New York city) were prospectively followed in an observational study using the custom Warrior Watch Study app, to collect weekly information about stress, symptoms and COVID-19 infection. Participants wore an Apple Watch for the duration of the study, measuring heart rate variability (HRV), a digital biomarker previously associated with infection in other settings, throughout the follow-up period.
The HRV data collected through the Apple Watch was characterized by a circadian pattern, with sparse sampling over a 24-hour period, and non-uniform timing across days and participants. These characteristics preclude us from using easily derived features (ie mean, maximum, CV etc) with Machine learning methods to develop a diagnostic tool. As such, suitable modelling of the non-uniform, sparsely sampled circadian rhythm data derived from wearable devices are an important step to advance the use of integrated wearable data for prediction of health outcomes.
To circumvent such limitations, we introduced the mixed-effects COSINOR model, where the daily circadian rhythm is express as a non-linear function with three rhythm characteristics: the rhythm-adjusted mean (MESOR), half the extent of variation within a cycle (amplitude), and an angle relating to the time at which peak values recur in each cycle (acrophase). The longitudinal changes in the circadian patterns can then be evaluated extending the COSINOR model to a mixed-effect model framework, allowing for random effects and interaction between COSINOR parameters and time-varying covariates. In this talk, we will discuss our model framework, boostrapped-based hypothesis testing and prediction approaches, as well as our evaluation of HRV measures as early biomarkers of COVID-19 diagnosis. To facilitate the future use of the mixed-effect COSINOR model, we implemented in an R package cosinoRmixedeffects.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Mayte Suarez-Farinas (Icahn School of Medicine at Mount Sinai, Estados Unidos) - sesion: Teorı́a de bordismo y acciones de grupos finitos zoom: https://us02web.zoom.us/j/89315141715 organizadores: - nombre: Andrés Ángel mail: ja.angel908@uniandes.edu.co - nombre: Ana González de los Santos mail: anagon@fing.edu.uy - nombre: Rita Jiménez Rolland mail: rita@im.unam.mx - nombre: Carlos Segovia mail: csegovia@matem.unam.mx charlas: - titulo: On the evenness conjecture for equivariant unitary bordism abstract:The evenness conjecture for equivariant unitary bordism states that these homology groups are free modules of even degree with respect to the unitary bordism ring. This conjecture is known to be true for all finite abelian groups and some semidirect products of abelian groups. In this talk I will talk about some progress done with the group of quaternions of order 8, together with an approach of Eric Samperton and Carlos Segovia to construct a free action on a surface which does not bound equivariantly. This last construction might provide a counterexample of the evenness conjecture in dimension 2.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Bernardo Uribe (Universidad del Norte, Colombia) - titulo: Extending free group action on surfaces abstract:We are interested in the question of when a free action of a finite group on a closed oriented surface extends to a non-necessarily free action on a 3-manifold. We show the answer to this question is affirmative for abelian, dihedral, symmetric and alternating groups. We present also the proof for finite Coxeter groups.
start: 2021-09-15T16:45 end: 2021-09-16T17:30 speaker: Carlos Segovia (Universidad Nacional Autónoma de México, México) - titulo: When does a free action of a finite group on a surface extend to a (possibly non-free) action on a 3-manifold? abstract:Dominguez and Segovia recently asked the question in the title, and showed that the answer is “always” for many examples of finite groups, including symmetric groups, alternating groups and abelian. Surprisingly, in joint work with Segovia, we have found the first examples of finite groups that admit free actions on surfaces that do NOT extend to actions on 3-manifolds. Even more surprisingly, these groups are already known in algebraic geometry as counterexamples to the Noether conjecture over the complex numbers. In this talk, I will explain how to find these groups, and, more generally, how to decide algorithmically if any fixed finite group admits a non-extending action.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Eric G. Samperton (University of Illinois, Estados Unidos) - titulo: The Loch Ness Monster as Homology Covers abstract:The Loch Ness Monster (LNM) is, up to homeomorphisms, the unique orientable, second countable, connected and Hausdorff surface of infinite genus and exactly one end. In this talk I would like to discuss some properties on the LNM. In particular, we note that LNM is the homology cover of most of the surfaces and also it is the derived cover of uniformizations of Riemann surfaces (with some few exceptions). This is a joint work, in progress, with Ara Basmajian.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Rubén A. Hidalgo (Universidad de la Frontera, Chile) - titulo: Diffeomorphisms of reducible three manifolds and bordisms of group actions on torus abstract:I first talk about the joint work with K. Mann on certain rigidity results on group actions on torus. In particular, we show that if the torus action on itself extends to a \(C^0\) action on a three manifold \(M\) that bounds the torus, then \(M\) is homeomorphic to the solid torus. This also leads to the first example of a smooth action on the torus via diffeomorphisms that isotopic to the identity that is nontrivial in the bordisms of group actions. Time permitting, I will also talk about certain finiteness results about classifying space of reducible three manifolds which came out of the cohomological aspect of the above project.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Sam Nariman (Purdue University, Estados Unidos) - titulo: Stolz' Positive scalar curvature surgery exact sequence and low dimensional group homology abstract:We will show how positive knowledge about the Baum-Connes Conjecture for a group, together with a Pontrjagyn character and knowledge about the conjugacy classes of finite subgroups and their low dimensional homology provide an estimation of the degree of non rigidity of positive scalar curvature metrics of spin high dimensional manifolds with the given fundamental group.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Noé Bárcenas (Universidad Nacional Autónoma de México, México) - titulo: Signature of manifold with single fixed point set of an abelian normal group abstract:In this talk we will report progress made in the task of reducing the computation of the signature of a (possibly non compact) oriented smooth dimensional manifold with an orientation preserving co-compact smooth proper action of a discrete group with a single non empty fixed point submanifold of fixed dimension with respect to the action of a (finite) abelian normal subgroup. The aim of this reduction is to give a formula for the signature in terms of the signature of a manifold with free action and the signature of a disc neighborhood of its fixed-point set with quasi-free action. A necessary task in this reduction is the generalization of Novikov’s additivity to this context.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Quitzeh Morales (Universidad Pedagógica de Oaxaca, México) - titulo: \(Z_p\)-bordism and the mod(\(p\))-Borsuk-Ulam Theorem abstract: |Crabb-Gonçalves-Libardi-Pergher classified for given integers \(m,n, \geq 1\) the bordism class of a closed smooth \(m\)-manifold \(X^m\) with a free smooth involution \(\tau\) with respect to the validity of the Borsuk-Ulam property that for every continuous map \(\varphi: X^m \to R^n\), there exists a point \(x \in X^m\) such that \(\varphi(x)= \varphi(\tau(x))\).
In this work together with Barbaresco-de Mattos-dos Santos-da Silva, we are considering the same problem for free \(Z_p\) action.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Alice Kimie Miwa Libardi (Universidad Estatal Paulista, Brasil) - sesion: Ecuaciones de evolución no-lineales y su Dinámica zoom: https://us02web.zoom.us/j/82032867591 organizadores: - nombre: Jaime Angulo Pava mail: angulo@ime.usp.br - nombre: Diego Chamorro mail: diego.chamorro@univ-evry.fr - nombre: Claudio Muñoz mail: cmunoz@dim.uchile.cl charlas: - titulo: Regularity criteria for weak solutions to the three-dimensional MHD system abstract: |In this talk we will first review various known regularity criteria and partial regularity theory for 3D incompressible Navier-Stokes equations. I will then present two generalizations of partial regularity theory of Caffarelli, Kohn and Nirenberg to the weak solutions of MHD equations. The first one is based on the framework of parabolic Morrey spaces. We will show parabolic Hölder regularity for the "suitable weak solutions" to the MHD system in small neighborhoods. This type of parabolic generalization using Morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory and makes it possible to weaken the hypotheses on the pressure. The second one is a regularity result relying on the notion of "dissipative solutions". By making use of the first result, we will show the regularity of the dissipative solutions to the MHD system with a weaker hypothesis on the pressure.
This is a joint work with Diego Chamorro (Université Paris-Saclay, site Evry).
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Jiao He (Université Paris-Saclay, Francia) - titulo: On the infinite energy weak solutions for the MHD equations abstract: |In this talk, we will consider the magneto-hydrodynamics (MHD) equations placed on the whole three-dimensional space. These equations write down as a nonlinear coupled system of two Navier-Stokes type equations, where the unknowns are the velocity field, the magnetic field and the pressure term. In the first part of this talk, within the framework of a kind of weighted L^2 spaces, we expose some new energy controls which allow us to prove the existence of global in time weak solutions. These solutions are also called the infinite energy weak solutions, in contrast with the classical theory of finite energy weak solutions in the L^2 space. The uniqueness of both finite energy and infinite energy weak solutions remains a very challenging open question. Thus, in the second part of this talk, we study a a priori condition, known as weak-strong uniqueness criterion, to ensure the uniqueness of the infinite energy weak solutions. This result is given in a fairly general multipliers type space, which contains some well-known functional spaces previously used to prove some weak-strong uniqueness criteria.
This is a joint work with Pedro Fernandez-Dalgo (Université Paris-Saclay).
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Oscar Jarrín (Universidad de las Américas, Ecuador) - titulo: Long-time asymptotics for a damped Navier-Stokes-Bardina model abstract:We consider finite energy solutions for a damped Navier-Stokes-Bardina’s model and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension. Furthermore, we examine the long-term behavior of solutions of the damped Navier-Stokes-Bardina’s equation in the energy space.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Fernando Cortez (Escuela Politécnica Nacional, Ecuador) - titulo: Compact embeddings of p-Sobolev-like cones of nuclear operators abstract:We prove that a cone of nuclear operators, whose eigenfunctions belong to a p-Sobolev space and that have finite total energy, is compactly embedded in the trace norm. This result is analogous to the classical Sobolev embeddings but at operators level. In the path we prove regularity properties for the density function of any operator living in the cone. Also, departing from Lieb-Thirring type conditions, we obtain some Gagliardo-Nirenberg inequalities. By using the compactness property, several free-energy functionals for operators are shown to have a minimizer. The entropy term of these free-energy functionals is generated by a Casimir-class function related to the eigenvalue problem of the Schrödinger operator.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Juan Mayorga (Yachay Tech, Ecuador) - titulo: Scattering for quadratic-type Schrödinger systems in dimension five without mass-resonance abstract:In this talk we will discuss the scattering of non-radial solutions in the energy space to coupled system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five without the mass-resonance condition. Our approach is based on the recent ideas introduced by Dodson and Murphy, which relies on an interaction Morawetz estimate.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Ademir Pastor (Universidade Estadual de Campinas, Brasil) - titulo: Stability of smooth periodic traveling waves in the Camassa-Holm equation abstract: |Smooth periodic travelling waves in the Camassa–Holm (CH) equation are revisited in this talk. We show that these periodic waves can be characterized in two different ways by using two different Hamiltonian structures. The standard formulation, common to the Korteweg–de Vries (KdV) equation, has several disadvantages, e.g., the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We explore the nonstandard formulation common to evolution equations of CH type and prove that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region of three parameters where the smooth periodic waves exist.
This is a joint work with Dmitry E. Pelinovsky (McMaster University), AnnaGeyer (Delft University of Technology) and Renan H. Martins (State University of Maringa).
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Fábio Natali (Universidade Estadual de Maringá, Brasil) - titulo: Spectral stability of monotone traveling fronts for reaction diffusion-degenerate Nagumo equations abstract:This talk addresses the spectral stability of monotone traveling front solutions for reaction diffusion equations where the reaction function is of Nagumo (or bistable) type and with diffusivities which are density dependent and degenerate at zero (one of the equilibrium points of the reaction). Spectral stability is understood as the property that the spectrum of the linearized operator around the wave, acting on an exponentially weighted space, is contained in the complex half plane with non-positive real part. The degenerate fronts studied in this paper travel with positive speed above a threshold value and connect the(diffusion-degenerate) zero state with the unstable equilibrium point of the reaction function. In this case, the degeneracy of the diffusion coefficient is responsible of the loss of hyperbolicity of the asymptotic coefficient matrices of the spectral problem at one of the end points, precluding the application of standard techniques to locate the essential spectrum. This difficulty is overcome with a suitable partition of the spectrum, a generalized convergence of operators technique, the analysis of singular (or Weyl) sequences and the use of energy estimates. The monotonicity of the fronts, as well as detailed descriptions of the decay structure of eigenfunctions on a case by case basis, are key ingredients to show that all traveling fronts under consideration are spectrally stable in a suitably chosen exponentially weighted L2 energy space.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Ramón G. Plaza (Universidad Nacional Autónoma de México, México) - titulo: Blow-up solutions of the intercritical inhomogeneous NLS equation abstract: |We consider the inhomogeneous nonlinear Schrödinger (INLS) equation $$i u_t +\Delta u+|x|^{-b}|u|^{2\sigma} u = 0, \,\,\, x\in \mathbb{R}^N,$$ with \(N\geq 3\) and \(0 \lt b \lt \min\{\frac{N}{2},2\}\). We focus on the intercritical case, where the scaling invariant Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma}\) satisfies \(0 \lt s_c \lt 1\). In this talk, for initial data in \(\dot H^{s_c}\cap \dot H^1\), we discuss the existence of blow-up solutions and also a lower bound for the blow-up rate in the radial and non-radial settings.
This is a joint work with Mykael Cardoso (Universidade Federal do Piauí, Brasil).
start: 2021-09-17T16:45 end: 2021-09-16T17:30 speaker: Luiz Gustavo Farah (Universidade Federal de Minas Gerais, Brasil). - titulo: A sufficient condition for asymptotic stability of kinks in general (1+1)-scalar field models abstract: |In this talk I will discuss stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models \(\partial_t2\phi -\partial_x2\phi + W'(\phi) = 0, \quad (t,x)\in\mathbb{R}\times\mathbb{R}\). The orbital stability of kinks under general assumptions on the potential \(W\) is a consequence of energy arguments. The main result I will present is the derivation of a simple and explicit sufficient condition on the potential \(W\) for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Applications of the criterion to the \(P(\phi)_2\) theories and the double sine-Gordon theory will be discussed.
This is a joint work with Y. Martel, C. Muñoz and H. Van Den Bosch.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Michal Kowalczyk (Universidad de Chile, Chile) - titulo: Korteweg de-Vries limit for the Fermi-Pasta-Ulam System abstract:In this talk, we are going to discuss dispersive properties for the Fermi-Pasta-Ulam (FPU) system with infinitely many oscillators. Precisely, we see that FPU systems are reformulated and their solutions satisfies Strichartz, local smoothing, and maximal function estimates in comparison with linear Korteweg–de Vries (KdV) flows. With these properties, we finally show that the infinite FPU system can be approximated by counter-propagating waves governed by the KdV equation as the lattice spacing approaches zero.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Chulkwang Kwak (Ehwa Womans University, Corea del Sur) - titulo: Spectral stability in the nonlinear Dirac equation with Soler type nonlinearity. abstract: |This talk concerns the nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space-dependent mass, known as the (generalized) Soler model. The equation has standing wave solutions for frequencies w in (0,m), where m is the mass in the Dirac operator. These standing waves are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations, but there are very few results in this direction.
The results that I will discuss concern the simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-self-adjoint operator with a particular block structure. I will present some partial results for the one-dimensional case, highlight the differences and similarities with the Schrödinger case, and discuss (a lot of) open problems.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Hanne Van Den Bosch (Universidad de Chile, Chile) - sesion: Combinatoria zoom: https://us02web.zoom.us/j/83421980657 organizadores: - nombre: Celina Miraglia Herrera de Figueiredo mail: celina@cos.ufrj.br - nombre: Flavia Bonomo mail: fbonomo@dc.uba.ar - nombre: Lucia Moura mail: lmoura@uottawa.ca charlas: - titulo: Complete colorings on circulant graphs and digraphs abstract: |A complete \(k\)-vertex-coloring of a graph \(G\) is a vertex-coloring of \(G\) using \(k\) colors such that for every pair of colors there is at least two incident vertices in \(G\) colored with this pair of colors. The \emph{chromatic} \(\chi(G)\) and \emph{achromatic} \(\alpha(G)\) numbers of \(G\) are the smallest and the largest number of colors in a complete proper \(k\)-vertex-coloring of \(G\), therefore \(\chi(G)\leq\alpha(G)\).
The dichromatic number and the diachromatic number, generalice the concepts of chromatic number and achromatic number. An acyclic \(k\)-vertex-coloring of a digraph \(D\) is vertex coloring using \(k\) colors such that \(D\) has no monochromatic cycles and a \emph{complete} \(k\)-vertex-coloring of a digraph \(D\) is a vertex coloring using \(k\) colors such that for every ordered pair \((i,j)\) of different colors, there is at least one arc \((u,v)\) such that \(u\) has color \(i\) and \(v\) has color \(j\). The dichromatic number \(dc(D)\) and diachromatic number \(dac(D)\) of \(D\) are the smallest and the largest number of colors in a complete proper \(k\)-vertex-coloring of \(D\), therefore \(dc(D)\leq dac(D)\).
We determine the achromatic and diachromatic numbers of some specific circulant graphs and digraphs and give general bounds for these two parameters on these graphs and digraphs. Also, we determine the achromatic index for circulant graphs of order \(q^2+q+1\) using projective planes.
Joint work with: Juan José Montellano-Ballesteros, Mika Olsen and Christian Rubio-Montiel.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Gabriela Araujo-Pardo (Universidad Nacional Autónoma de México, México) - titulo: Fault Localization via Combinatorial Testing abstract: |In this talk, we explore combinatorial arrays that are useful in identifying faulty interactions in complex engineered systems. A separating hash family \({\sf SHF}_\lambda(N; k,v,\{w_1,\dots,w_s\})\) is an \(N \times k\) array on \(v\) symbols, with the property that no matter how disjoint sets \(C_1, \dots, C_s\) of columns with \(|C_i| = w_i\) are chosen, there are at least \(\lambda\) rows in which, for every \(1 \leq i < j \leq s\), no entry in a column of \(C_i\) equals that in a column of \(C_j\). (That is, there are \(\lambda\) rows in which sets \(\{C_1,\dots,C_s\}\) are separated) Separating hash families have numerous applications in combinatorial cryptography and in the construction of various combinatorial arrays; typically, one only considers whether two symbols are the same or different. We instead employ symbols that have algebraic significance.
We consider an \({\sf SHF}_\lambda(N; k,q^s,\{w_1,\dots,w_s\})\) whose symbols are column vectors from \({\mathbb F}_q^s\). The entry in row \(r\) and column \(c\) of the \({\sf SHF}\) is denoted by \({\bf v}_{r,c}\). Suppose that \(C_1, \dots, C_s\) is a set of disjoint sets of columns. Row \(r\) is covering for \(\{C_1, \dots, C_s\}\) if, whenever we choose \(s\) columns \(\{ \gamma_i \in C_i : 1 \leq i \leq s\}\), the \(s \times s\) matrix \([ {\bf v}_{r,\gamma_1} \cdots {\bf v}_{r,\gamma_s} ]\) is nonsingular over \({\mathbb F}_q\). Then the \({\sf SHF}_\lambda(N; k,q^s,\{w_1,\dots,w_s\})\) is covering if, for every way to choose \(\{C_1, \dots, C_s\}\), there are at least \(\lambda\) covering rows.
We establish that covering separating hash families of type \(1^t d^1\) give an effective construction for detecting arrays, which are useful in screening complex systems to find interactions among \(t\) or fewer factors without being masked by \(d\) or fewer other interactions. This connection easily accommodates outlier and missing responses in the screening. We explore asymptotic existence results and explicit constructions using finite geometries for covering separating hash families. We develop randomized and derandomized construction algorithms and discuss consequences for detecting arrays. This is joint work with Violet R. Syrotiuk (ASU).
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Charles J. Colbourn (Arizona State University, Estados Unidos) - titulo: On the \(\Delta\)-interval and \(\Delta\)-convexity numbers of graphs and graph products abstract: |Given a graph \(G\) and a set \(S \subseteq V(G)\), the \(\Delta\)-interval of \(S\), \([S]_{\Delta}\), is the set formed by the vertices of \(S\) and every \(w \in V(G)\) forming a triangle with two vertices of \(S\). If \([S]_\Delta = S\), then \(S\) is \(\Delta\)-convex of \(G\); if \([S]_{\Delta} = V(G)\), then \(S\) is a \(\Delta\)-interval set of \(G\). The \(\Delta\)-interval number of \(G\) is the minimum cardinality of a \(\Delta\)-interval set and the \(\Delta\)-convexity number of \(G\) is the maximum cardinality of a proper \(\Delta\)-convex subset of \(V(G)\). In this work, we show that the problem of computing the \(\Delta\)-convexity number is {\sf W}[1]-hard and {\sf NP}-hard to approximate within a factor \(O(n^{1-\varepsilon})\) for any constant \(\varepsilon > 0\) even for graphs with diameter \(2\) and that the problem of computing the \(\Delta\)-interval number is {\sf NP}-complete for general graphs. For the positive side, we present characterizations that lead to polynomial-time algorithms for computing the \(\Delta\)-convexity number of chordal graphs and for computing the \(\Delta\)-interval number of block graphs. We also present results on the \(\Delta\)-hull, \(\Delta\)-interval and \(\Delta\)-convexity numbers concerning the three standard graph products, namely, the Cartesian, strong and lexicographic products, in function of these and well-studied parameters of the operands.
Joint work with: Bijo S. Anand, Prasanth G. Narasimha-Shenoi, and Sabeer S. Ramla.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Mitre Dourado (Universidade Federal do Rio de Janeiro, Brasil) - titulo: Unavoidable patterns abstract:Ramsey's theorem states that, for any integer \(t\), if \(n\) is a sufficiently large integer, then every \(2\)-edge-coloring of a complete graph \(K_n\) contains a monochromatic \(K_t\). On the other hand, Turán's theorem says that, if a graph has sufficiently many edges, then it contains a \(K_t\) as a subgraph. In relation to these two types of problems, we study which \(2\)-colored patterns are forced to appear in differently saturated \(2\)-edge-colorings of the complete graph \(K_n\) provided \(n\) is large enough. This is a joint work with Yair Caro and Amanda Montejano.
start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Adriana Hansberg (Universidad Nacional Autónoma de México, México) - titulo: Approximating graph eigenvalues using tree decompositions abstract: |Spectral graph theory aims to extract structural information about graphs from spectra of different types of matrices associated with them, such as the adjacency matrix, Laplacian matrices, and many others. A basic step in any such application consists of computing or approximating the spectrum or some prescribed subset of eigenvalues. One strategy for this is to compute diagonal matrices that are congruent to the original matrices, which can be done quite efficiently if we have a graph decompositions such as the tree decomposition with bounded width. To be precise, let \(M=(m_{ij})\) be a symmetric matrix of order \(n\) whose elements lie in an arbitrary field \(\mathbb{F}\), and let \(G\) be the graph with vertex set \(\{1,\ldots,n\}\) such that distinct vertices \(i\) and \(j\) are adjacent if and only if \(m_{ij} \neq 0\). We present an algorithm that finds a diagonal matrix that is congruent to \(M\) in time \(O(k^2 n)\), provided that we are given a suitable tree decomposition of \(G\) with width \(k\). In addition to the above applications, this allows one to compute the determinant and to find the rank of a symmetric matrix in time \(O(k^2 n)\).
This is joint work with M. FŸrer (Pennsylvannia State University) and Vilmar Trevisan (Universidade Federal do Rio Grande do Sul).
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Carlos Hoppen (Universidade Federal do Rio Grande do Sul, Brasil) - titulo: Fault tolerance in cryptography using cover-free families abstract: |Cover-free families (CFFs) have been investigated under different names and as a solution to many problems related to combinatorial group testing. A \(d\)-cover-free family \(d\)-CFF\((t, n)\) is a set system with n subsets of a \(t\)-set, where the union of any d subsets does not contain any other. A \(d\)-CFF\((t, n)\) allows for the identification of up to \(d\) defective elements in a set of \(n\) elements by performing only \(t\) tests (typically \(t \ll n\)).
We explore different aspects of cover-free families in order to achieve fault tolerance in cryptography. For instance, while CFFs are used as a solution to many problems in this area, we note that some of those problems require CFFs with increasing \(n\). In this context, we investigate new infinite families and their constructions in order to better approach fault tolerance in cryptography. This is joint work with Lucia Moura.
[1] IDALINO, T. B.; MOURA, L., Nested cover-free families for unbounded fault-tolerant aggregate signatures. Theoretical Computer Science, 854 (2021), 116--130.
[2] IDALINO, T. B.; MOURA, L., Embedding cover-free families and cryptographical applications. Advances in Mathematics of Communications, 13 (2019), 629--643.
An \((n,k,\lambda)\) perfect sequence covering array is a subset of the \(n!\) permutations of the sequence \((1, 2, \dots, n)\) whose elements collectively contain each ordered \(k\)-subsequence exactly \(\lambda\) times. The central question is: for given \(n\) and \(k\), what is the smallest value of \(\lambda\) (denoted \(g(n,k)\)) for which such a configuration exists? We interpret the sequences of a perfect sequence covering array as elements of the symmetric group \(S_n\), and constrain its structure to be a union of cosets of a prescribed subgroup of \(S_n\). By adapting a search algorithm due to Mathon and van Trung for finding spreads, we obtain highly structured examples of perfect sequence covering arrays. In particular, we determine that \(g(6,3) = g(7,3) = g(7,4) = 2\) and that \(g(8,3)\) is either 2 or 3.
This is joint work with Jingzhou Na.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Jonathan Jedwab (Simon Fraser University, Canada) - titulo: The status of the \((r, l)\) well-covered graph sandwich problem abstract: |An \((r, l)\)-partition of a graph \(G\) is a partition of its vertex set into \(r\) independent sets and \(l\) cliques. A graph is \((r,l)\) if it admits an \((r, l)\)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is \((r,l)\)-well-covered if it is both \((r,l)\) and well-covered. We have proved the full classification of the decision problem \((r,l)\)-Well-Covered Graph problem, where we are given a graph \(G\), and the question is whether \(G\) is an \((r, l)\)-well-covered graph. We have shown that this problem is polynomial only in the cases \((0, 1)\), \((0, 2)\), \((1, 0)\), \((1, 1)\), \((1, 2)\), and \((2, 0)\) and hard otherwise. The Sandwich Problem for the Property \(\pi\) has as an instance a pair of graphs \(G1 = (V, E_1)\) and \(G2 = (V, E_2)\) with \(E_1 \subset E_2\), plus the question whether there is a graph \(G=(V,E)\) with \(E_1 \subset E\subset E_2\), such that \(G\) has the property \(\pi\). When a recognition problem for a property \(\pi\) is hard, we can consider the sets \(E_1 = E_2\) to obtain that the sandwich problem for the property \(\pi\) reduces to the recognition and so also hard. Hence, the only cases where the \((r, l)\)-well-covered sandwich problem can be no longer hard are in the \(6\) polynomial cases. In this talk we prove that \((r, l)\)-well-covered sandwich problem is polynomial in the cases that \((r, l) = (0, 1), (1, 0), (1, 1)\) or \((0, 2)\), and NP-complete if we consider the property of being \((1, 2)\)-well-covered.
This work was done with the collaboration of Sancrey Rodrigues Alves (FAETEC), Fernanda Couto (UFRRJ), Luerbio Faria (UERJ), Sylvain Gravier (CNRS), and Uéverton S. Souza (UFF).
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Sulamita Klein (Universidade Federal do Rio de Janeiro, Brasil) - titulo: Counting Segment, Rays, Lines in Quasimetric spaces abstract:In the Euclidean plane, it is a well-known result that a set of n points defines exactly \(n(n-1)/2\) distinct segments, at least \(2(n-1)-1\) different rays and, when the points are not collinear, at least \(n\) different lines. Segments, rays and lines can be define in any quasimetric space by using its quasimetric. In this talk we survey recent results concerning the generalization of the above mentioned properties of the Euclidean plane, specially those of lines, to finite quasimetric spaces, mainly those defined by connected graphs and strongly connected digraphs.
start: 2021-09-14T17:30 end: 2021-09-13T18:15 speaker: Martín Matamala (Universidad de Chile, Chile) - titulo: Uniformly Optimally-Reliable Graphs abstract: |The reliability polynomial of a simple graph G represents the probability that G will remain connected given a fixed probability \(1-p\) of each edge failing. A graph \(G\) is uniformly most reliable if its reliability polynomial is greater than or equal to the reliability polynomial of all other graphs with the same number of nodes and edges for all \(p \in [0, 1]\).
Which is the most-reliable graph with n nodes and m edges?
This problem has several variants, according to the notion of optimality (in a local or uniform sense), failure type (either nodes or edges), or reliability model (all-terminal connectedness, source-terminal or general multi-terminal setting).
This presentation shows a summary of the multiple proposals to attack the problem, together with recent trends and important conjectures proposed decades ago which require further research.
Particularly, we will present recently proved conjectures regarding UMGR (Uniformly Most-Reliable Graphs) as well as new evaluation techniques for this property.
Keywords: uniformly most-reliable graph, uniformly least-reliable graph, all-terminal reliability, source-terminal reliability, failure-type, graph theory.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Franco Robledo (Universidad de la Republica, Uruguay) - titulo: Circularly compatible ones, \(D\)-circularity, and proper circular-arc bigraphs abstract:In 1969, Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented adjacency matrix has the circularly compatible ones property. These results led to the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as remarked there, this work did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones property in arbitrary matrices (i.e., not restricted to augmented adjacency matrices only). We solve these problems. More precisely, we give a minimal forbidden submatrix characterization for the circularly compatible ones property in arbitrary matrices and a linear-time recognition algorithm for the same property. We derive these results from analogous ones for the related \(D\)-circular property. Interestingly, these results lead to a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs, solving a problem first posed by Basu et al. [J. Graph Theory, 73 (2013), pp. 361--376]. Our findings generalize some known results about \(D\)-interval hypergraphs and proper interval bigraphs.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Martín Safe, (Universidad Nacional del Sur, Argentina) - titulo: Active clustering abstract: |Given a set (known to us), and a partition of this set, unknown to us, consider the following task. We have to determine the partition by making successive queries about pairs of elements, each time querying whether the pair belong to the same class or not. Later queries are allowed to depend on the outcome of earlier queries. There is a natural way to assign a graph to each step of the querying process, by starting with an edgeless graph where each vertex represents an element of the set, and after queries identifying vertices for a yes-answer and adding edges for no-answers. We prove the at first sight surprising result that the partitioning algorithms reaching the minimal average number of queries are exactly those that keep these graphs chordal. (The average is taken over all possible partitions of the base set.) The optimal algorithms even share the same distribution on the number of queries, which we characterize. We also analyse a related model where the number of partition classes is fixed, and each element of the base set chooses its partition class randomly.
This is joint work with Élie de Panafieu, Quentin Lutz and Alex Scott.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Maya Stein (Universidad de Chile, Chile) - sesion: Análisis Numérico zoom: https://us02web.zoom.us/j/88382162185 organizadores: - nombre: Gabriel Acosta mail: gacosta@dm.uba.ar - nombre: Pedro Morin mail: pmorin@fiq.unl.edu.ar - nombre: Rodolfo Rodríguez mail: rodolfo@ing-mat.udec.cl charlas: - titulo: Aproximación unificada del problema acoplado de Stokes-Darcy abstract: |En esta charla nos abocaremos al análisis y la resolución numérica, por elementos finitos mixtos, del problema acoplado de Stokes-Darcy en el plano. Introduciremos una formulación modificada del problema con el propósito de permitir el uso de la misma familia de elementos, en el sector del dominio gobernado por la ecuación de Stokes y en la porción del dominio gobernada por la ecuación de Darcy.
En primer lugar consideraremos el caso de que el dominio sea poligonal, donde presentaremos resultados tanto teóricos como numéricos de la resolución del problema utilizando MINI-elements, los cuales son uno de los más sencillos de implementar. También, aprovechando las ventajas de nuestra formulación, mostraremos la buena performance de la aplicación de otros elementos como los P2-P1 o P1-P1 estabilizados.
Luego trataremos el problema en dominios curvos, incluso con la posibilidad de tener interfase curva, haciendo uso de triángulos curvos. Con este enfoque obtendremos también estimaciones de error de orden óptimo, extendiendo así los resultados obtenidos para el caso poligonal. Por último presentaremos experimentos numéricos que confirman la buena performance del método propuesto.
Trabajo en colaboración con María Lorena Stockdale. Departamento de Matematica, FCEyN, Universidad de Buenos Aires.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Gabriela Armentano (Universidad de Buenos Aires, Argentina) - titulo: 'Transversally-Enriched Pipe Element Method: an efficient approach for computational hemodynamics' abstract:The development of novel numerical and computational strategies for the simulation of blood flow in complex patient-specific vasculatures has gained strong attention in the scientific community aiming both theoretical and applied research. The need for fast, yet accurate methods is at the core of any translational research enterprise. In this talk, we will introduce a middle fidelity numerical approach to tackle the demanding blood-flow simulation in deformable arteries. This strategy can be placed in-between oversimplified (low-fidelity) reduced 1D models and expensive (high-fidelity) full 3D models. The main feature of the method is the discretization of the pipe-like domain in which the fluid flow problem is addressed. This pipe-oriented discretization is exploited by placing a differentiated polynomial approximation for cross-sectional and longitudinal directions. We will show the basic idea of the method, a numerical study of its convergence properties and applications for patient-specific blood flow modeling in large branching arterial geometries.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Pablo Blanco (Laboratório Nacional de Computação Científica, Brasil) joint with Alonso M. Alvarez, Raúl A. Feijóo - titulo: Local error estimates for nonlocal problems abstract: |The integral fractional Laplacian of order \(s \in (0,1)\) is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions.
In this talk we shall discuss regularity of solutions on bounded Lipschitz domains. For finite element discretizations, we derive local error estimates in the \(H^s\)-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples.
The talk is based on joint work with Dmitriy Leykekhman (University of Connecticut) and Ricardo Nochetto (University of Maryland).
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Juan Pablo Borthagaray (Universidad de la República, Uruguay) - titulo: A priori and a posteriori error analysis of a momentum conservative mixed-FEM for the stationary Navier-Stokes problem abstract: |In this talk we present a new conforming mixed finite element method for the Navier-Stokes problem posed on non-standard Banach spaces, where a pseudostress tensor and the velocity are the main unknowns of the system. The associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the pseudostress and discontinuous piecewise polynomials of degree k for the velocity. Next, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed scheme.
Joint work with S. Caucao (Universidad Católica de la Santísima Concepción, Chile), C. Garcia (Pontificia Universidad Católica de Chile), R. Oyarzúa (Universidad del Bío Bío, Chile) and S. Villa (Universidad del Bío Bío, Chile).
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Jessika Camaño (Universidad Católica de la Santísima Concepción, Concepción, Chile) - titulo: A fully discrete adaptive scheme for parabolic equations abstract: |We present an adaptive algorithm for solving linear parabolic equations using hierarchical B-splines and the implicit Euler method for the spatial and time discretizations, respectively.
Our development improves upon one from 2018 by Gaspoz and collaborators, where fully discrete adaptive schemes have been analyzed within the framework of classical finite elements. Our approach is based on an a posteriori error estimation that essentially consists of four indicators: a time and a consistency error indicator that dictate the time-step size adaptation, and coarsening and a space error indicator that are used to obtain suitably adapted hierarchical meshes (at different time steps). Even though we use hierarchical B-splines for the space discretization, a straightforward generalization to other methods, such as FEM, is possible. The algorithm is guaranteed to reach the final time within a finite number of operations, and keep the space-time error below a prescribed tolerance. Some numerical tests document the practical performance of the proposed adaptive algorithm.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Eduardo Garau (Universidad Nacional del Litoral, Santa Fe, Argentina) - titulo: LSD for multiscale problems abstract: |The Localized Spectral Decomposition finite element method is based on a hybrid formulation of elliptic partial differential equations, that is then transformed via a FETI-like space decomposition. Such decomposition make the formulation embarrassingly parallel and efficient, in particular in the presence of multiscale coefficients. It differs from most of the methods out there since it requires solution's minimum regularity. The major novelty is that we base our decomposition in local spectral problems, resulting in methods that are robust with respect to high contrast coefficients.
This is a joint work with Marcus Sarkis, of WPI (Worcester Polytechnic Institute).
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Alexandre Madureira (Laboratório Nacional de Computação Científica, Brasil) - titulo: Virtual Element Spectral Analysis for the Transmission Eigenvalue Problem. abstract:The aim of this talk is to analyze a C1 Virtual Element Method (VEM) on polygonal meshes for solving a quadratic and non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained. Numerical experiments will be provided to verify the theoretical error estimates.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: David Mora (Universidad del Bio-Bio, Concepción, Chile) - titulo: Analysis and approximation of fluids under singular forcing abstract:We present a well-posedness theory for Newtonian and some non-Newtonian fluids under singular forcing in Lipschitz domains and in convex polytopes. The main idea, that allows us to deal with such forces, is that we study the fluid problems in suitably weighted Sobolev spaces. We develop an a priori approximation theory, which requires the development of the stability of the Stokes projection over weighted spaces. We conclude by presenting existence results for a singular Bousinessq system and, in the case that the forcing is a linear combination of Dirac deltas, we also present an a posteriori error estimator.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Enrique Otárola (Universidad Técnica Federico Santamaría, Valparaíso, Chile) - sesion: Análisis Funcional y Geometría zoom: https://us02web.zoom.us/j/84041238615 organizadores: - nombre: Esteban Andruchow mail: eandruch@ungs.edu.ar - nombre: Pedro Massey mail: massey@mate.unlp.edu.ar - nombre: Lázaro Recht mail: recht@usb.ve charlas: - titulo: Mínimos locales de problemas tipo Procusto en la variedad de matrices positivas abstract: |Sea \(\mathcal{M}_d(\mathbb{C})\) el espacio de matrices (cuadradas) de dimensión \(d\) y \(\mathcal{X}\subset \mathcal{M}_d(\mathbb{C})\). Consideremos una matriz \(A\in\mathcal{M}_d(\mathbb{C})\) (fija) y una métrica en \(\mathcal{M}_d(\mathbb{C})\) dada por una distancia \(\rm {\textbf d}\).
Un típico problema de aproximación de matrices (o de tipo Procusto) es estudiar la distancia mínima $$\rm {\textbf d}(A,\mathcal{X}):= \inf\{ \rm {\textbf d}(A,C):\,C \in \mathcal{X}\}\,,$$ y en caso de que se alcance, estudiar el conjunto de mejores aproximantes de \(A\) en \(\mathcal{X}\) $$\mathcal{A}^{\rm op}(A,\mathcal{X}) =\{C\in\mathcal{X}:\,\rm {\textbf d}(A,C)= \rm {\textbf d}(A,\mathcal{X})\}\,.$$ Algunas de las elecciones clásicas de \(\mathcal{X}\subset \mathcal{M}_d(\mathbb{C})\) son las matrices autoadjuntas, las semidefinidas positivas, los proyectores ortogonales, etc, y la métrica suele ser la inducida por la norma Frobenius, pero también podría provenir de cualquier otra norma, por ejemplo, de alguna que sea unitariamente invariante (nui).
El problema del que nos ocuparemos en esta charla es el siguiente: dada \(N\) una nui (estrictamente convexa) en \(\mathcal{M}_d(\mathbb{C})\) definimos en el cono de matrices positivas \(\mathcal{P}_d(\mathbb{C})\) la distancia $$ {\bf{d}}_N(A,B):=N(\log(A^{-1/2} B A^{-1/2})) \quad \text{para } A,B\in\mathcal{P}_d(\mathbb{C}). $$ Entonces, si fijamos \(A,B\in \mathcal{P}_d(\mathbb{C})\) podemos considerar $$\mathcal{X}=\mathcal{O}_B= \{ UBU^* : \, U\quad\text{es unitaria}\}\,.$$ Luego, el problema de Procusto asociado es el de estudiar la distancia $$ \displaystyle{\bf{d}}_N(A,\mathcal{O}_B) =\inf_{C\in\mathcal{O}_B} {\bf{d}}_N(A,C)$$ y (en caso de ser posible) los mejores aproximantes de \(A\) en \(\mathcal{O}_B\). En 2019, Bhatia y Congedo probaron que esa distancia se alcanza en matrices de \(\mathcal{O}_B\) que conmutan con \(A\) Como \(\mathcal{O}_B\) es un espacio métrico con la métrica inducida por la norma usual de operadores, lo que proponemos en esta charla es estudiar los minimizadores globales la función \(F_{(N,A,B)}= F_N:\mathcal{O}_B \to \mathbb{R}_{>0}\) dada por $$F_N(C)=N (\log (A^{-1/2}CA^{-1/2}))$$ para \(C\in \mathcal{O}_B\) En particular, vamos a dar una caracterización espectral de los minimizadores locales de \(F_N\) en \(\mathcal{O}_B\) (cuando \(N\) es una nui estrictamente convexa) utilizando técnicas geométricas aplicadas al caso de igualdad en la desigualdad de Lidskii (multiplicativa) y probaremos que los minimizadores locales son globales, independientemente de la nui estrictamente convexa elegida. La charla está basada en un trabajo en co-autoría con Pablo Calderón y Mariano Ruiz.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Noelia Belén Rios (Universidad Nacional de La Plata, Argentina) - titulo: On \(\lambda\)-Rings of Pseudo-differential Operators abstract: |The theory of \(\lambda\) Rings goes back to the work of Grothendieck on Chern classes in algebraic topology, it is a suitable axiomatization of the algebraic properties of exterior powers operations on vector bundles; \(\lambda\) rings were also used by Atiyah and coworkers in the study of representations of groups and \(K\) Theory. During this talk we will present recent results on the \(\lambda\) ring structure in algebras of pseudo-differential operators and their use in index theory.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Alexander Cardona (Universidad de los Andes, Colombia) - titulo: El cuarto problema de Hilbert para analistas abstract: |El cuarto problema de Hilbert pide construir y estudiar las distancias continuas (aunque no necesariamente simétricas) en abiertos convexos de espacios proyectivos para las cuales las rectas son lineas geodésicas. En dimensión dos, y para distancias simétricas, la solución de Busemann y Pogorelov es de una maravillosa simplicidad. Sin embargo, en dimensiones mayores a dos y para distancias no simétricas podemos seguir considerando el problema como abierto. Después de una breve reseña de la solución bidimensional de Busemann y Pogorelov y de varios resultados parciales en dimensión superior, la charla se centrará en la formulación de problemas relacionados a la transformada de Fourier y a otras transformadas integrales que llevarían a una solución o a un mejor entendimiento del problema de Hilbert.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Juan Carlos Alvarez-Paiva (Université Lille 1, Francia) - titulo: Productos de operadores positivos abstract: |La charla se basa en un trabajo en colaboración con Maximiliano Contino, Michael Dritschel y Stefania Marcantognini sobre el conjunto de operadores en un espacio de Hilbert separable que pueden escribirse como el producto de dos operadores lineales, acotados y positivos. En espacios de dimensión finita, es fácil ver que un operador es el producto de dos operadores positivos si y sólo si es similar a un operador positivo, es decir si y sólo si es un operador escalar con espectro positivo. La estructura de este conjunto en espacios de dimensión infinita es mucho más rica y compleja. La factorization de un elemento del conjunto no es única pero existen factorizaciones distinguidas. La pertenencia a este conjunto se vincula con la cuasi-similaridad y cuasi afinidad a un operador positivo aunque no es equivalente. Estudiamos también las propiedades espectrales de los elementos del conjunto y describimos varios ejemplos.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Alejandra Maestripieri (Universidad de Buenos Aires, Argentina) - titulo: Diseño óptimo de multicompletaciones con restricciones de norma abstract: |Consideremos una sucesión finita de números reales positivos \(\alpha=(\alpha_i)_{i=1}^n\) y una sucesión de números enteros positivos \(\mathbf d=(d_j)_{j=1}^m\), ambas ordenadas en forma no-creciente.
Un \((\alpha,\mathbf d)-\) diseño es una familia \(\Phi=(\mathcal F_j)_{j=1}^m\) tal que: \(\mathcal F_j=\{f_{ij}\}_{i=1}^n \in (\mathbb C^{d_j})^n\) de forma que se verifican las restricciones $$\sum_{j=1}^m\|f_{ij}\|^2=\alpha_i\,,\ i=1,\ldots,n.$$ Denotaremos con \(\mathcal D(\alpha,\mathbf d)\) al conjunto de todos los \((\alpha,\mathbf d)-\) diseños.
Sea \(\Phi^0 =(\mathcal F^0_j)_{j=1}^m\) tal que \(\mathcal F^0_j=\{f^0_{ij}\}_{i=1}^k\in (\mathbb C^{d_j})^k\) con \(j=1,\ldots,m\). Una \((\alpha,\mathbf d)-\){ \it multicompletación} de\(\Phi^0\) es $$(\Phi^0,\Phi)=(\mathcal F^0_j,\mathcal F_j)_{j=1}^m \,\text{ con }\, \Phi\in \mathcal D(\alpha,\mathbf d)\,,$$ donde \((\mathcal F^0_j, \mathcal F_j)\in (\mathbb C^{d_j})^{k+n}\), para \(j=1, \ldots, m\). Dadas \((\Phi^0,\Phi)\) una \((\alpha,\mathbf d)-\) multicompletación y una función \(\varphi:\mathbb R_{\geq 0}\to \mathbb R_{\geq 0}\) estrictamente convexa, consideramos el potencial conjunto inducido por \(\varphi\), dado por: $$\Psi_{\varphi}(\Phi)= \rm P_{\varphi}(\Phi^0,\Phi)=\sum_{j=1}^m \text{tr}(\varphi[S_{(\mathcal F^0_j , \mathcal F_j)}]),$$ donde \(S_{(\mathcal F^0_j, \mathcal F_j)}=S_{\mathcal F^0_j}+S_{ \mathcal F_j}\) denota el operador de marco de \((\mathcal F^0_j, \mathcal F_j)\in (\mathbb C^{d_j})^{k+n}\), para \(j=1, \ldots, m\). Es bien sabido que los mínimos de potenciales convexos (bajo restricciones en las normas de los vectores) dan lugar a sistemas de reconstrucción más estables: cuanto menor es el potencial, más estable es el sistema.
En esta charla consideraremos el problema de la existencia de \((\alpha,\mathbf d)-\) multicompletaciones \((\Phi^0,\Phi^{\text{op}})\) óptimas dentro de la clase de todas las \((\alpha,\mathbf d)-\) multicompletaciones, es decir, tales que $$\rm P_{\varphi}(\Phi^0,\Phi^{\text{op}})\leq \rm P_{\varphi}(\Phi^0,\Phi),$$ para toda \((\alpha,\mathbf d)-\) multicompletación \((\Phi^0,\Phi)\) y para toda \(\varphi\).
Si \(m=1\) y \(\mathcal F_1^0\) es una sucesión inicial fija, entonces el problema anterior se reduce a hallar las completaciones óptimas de \(\mathcal F_1^0\) con normas predeterminadas por \(\alpha\) (este caso fue probado por P. Massey, N. Ríos y D. Stojanoff en 2018), que a su vez contiene el problema de diseño óptimo con normas predeterminadas i.e. \(\mathcal F_1^0=\{0\}\) (probado por M.B; P. Massey, M. Ruiz y D. Stojanoff en 2020). La charla está basada en un trabajo en co-autoría con P. Massey, M. Ruiz y D. Stojanoff.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: María José Benac (Universidad Nacional de Santiago del Estero, Argentina) - titulo: From closures of unitary group actions to groupoid actions abstract: |The set of all normal operators on a Hilbert space is naturally acted on both by the group of unitary operators and by the groupoid of partial isometries. We show that the norm closures of these group orbits are related to the above groupoid orbits, and this relation allows one to study the differentiable structures of these orbit closures. Some of the earlier results on closed unitary orbits are recovered in the natural degree of generality in our framework. This presentation is based on joint work with Gabriel Larotonda.
start: 2021-09-15T15:00 end: 2021-09-16T15:45 speaker: Daniel Beltita (Institute of Mathematics of the Romanian Academy, Rumania) - titulo: A nonlocal Jacobian equation? abstract: |We study an operator that assigns to each function \(u:\mathbb{R}^d\to\mathbb{R}\) a mapping \(G_u:\mathbb{R}^d \to C_*(\mathbb{R}^d)\), $$G_u(x)(h) := u(x+h)-u(x)\;\forall h \in \mathbb{R}^d.$$
This map \(G_u(x)\) has some similarities with the gradient map \(\nabla u(x)\), which is a central object of study in the theory of the Monge-Ampère equation and Jacobian equations in general. The image of the map \(G_u\) will be, in general, a \(d\)-dimensional submanifold inside the Banach space \(C_*(\mathbb{R}^d)\) (the space of continuous, bounded functions which vanish at the origin). Our goal is to find a relation, at least for some broad class of functions \(u\), between the oscillation of the function \(u\) in a compact domain \(D\) and the \(d\)-dimensional measure of the set \(G_u(D)\). Such a relation would be analogous to Aleksandrov's estimate for convex functions, a fundamental estimate in the theory of elliptic equations which can be traced back to the reverse Blaschke-Santaló inequality. The validity of an integro-differential version of this estimate would have significant implications for the study of nonlinear integro-differential equations. In this talk I will review this background and discuss some preliminary results about the map \(G_u\). This is work in progress.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Nestor Guillen (Texas State University, Estados Unidos) - titulo: Constant scalar curvature, scalar flat, and Einstein metrics abstract: | Let \((M^n,g)\) be a closed Riemannian manifold of dimension \(n\). Then:In this talk, I will start by explaining the nonhamiltonian nature of nonholonomic systems, and then we will study the "hamiltonization problem" from a geometric standpoint. By making use of symmetries and suitable first integrals of the system, we will explicitly define a new bracket on the reduced space codifying the nonholonomic dynamics that, in many examples, is a genuine Poisson bracket making the system hamiltonian (in general, the new bracket carries an almost symplectic foliation determined by the first integrals). This is a joint work with Luis Yapu-Quispe.
start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Paula Balseiro (Universidad Federal Fluminense, Brasil) - titulo: From retraction maps to geometric integrators for optimal control problems abstract: |Retraction maps are used in many research fields such as approximation of trajectories of differential equations, optimization theory, interpolation theory etc. In this talk we will review the concept of retraction map in differentiable manifolds to generalize it to obtain an extended retraction map from the tangent bundle of the configuration manifold to two copies of this manifold. After suitably lifting the new retraction map to the cotangent bundle, the typical phase space for Hamiltonian mechanical systems, we will be able to define geometric integrators for optimal control problems. This is a joint work with David Martín de Diego.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: María Barbero Liñán (Universidad Politécnica de Madrid, España) - titulo: Relación entre la teoría de campos de Chern-Simons y relatividad general en dimensión \(3\), desde el punto de vista de los problemas variacionales de Griffiths abstract: |La teoría de campos de Chern-Simons es una teoría de campos topológica, y puede formularse sobre cualquier fibrado principal cuyo grupo de estructura admita una forma bilineal invariante. Consideremos un fibrado principal \(\pi:P\to M\) con grupo de estructura \(G\) y sea \(K\subset G\) un subgrupo; tomemos la familia de problemas variacionales de Chern-Simons sobre todos los subfibrados de \(P\) con fibra \(K\). En la presente charla explicaremos cómo esta familia de problemas variacionales puede codificarse mediante un único principio variacional de tipo Griffiths. Finalmente, utilizaremos el problema variacional construído para entender desde un punto de vista geométrico la correspondencia entre gravedad en dimensión \(2+1\) y la teoría de Chern-Simons.
start: 2021-09-16T17:30 end: 2021-09-17T18:15 speaker: Santiago Capriotti (Universidad Nacional del Sur, Argentina) - titulo: Lie group's exponential curves and the Hamilton-Jacobi theory abstract: |In this talk we present an extended version of the Hamilton-Jacobi equation (HJE), valid for general dynamical systems defined by vector fields (not only by the Hamiltonian ones), and a result which ensures that, if we have a complete solution of the HJE for a given dynamical system, inside a certain subclass of systems, then such a system can be integrated up to quadratures. Then we apply this result to show that the exponential curves \(\exp\left(\eta\,t\right)\) of any Lie group, which are the integral curves of the left invariant vector fields in the group, can be constructed up to quadratures (unless for certain elements \(\eta\) inside its corresponding Lie algebra). This gives rise to an alternative concrete expression of \(\exp\left(\eta\,t\right)\), different to those that appears in the literature for matrix Lie groups.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Sergio Grillo (Centro Atómico Bariloche, Argentina) - titulo: Null hyperpolygons and quasi-parabolic Higgs bundles abstract: |Hyperpolygons spaces are a family of hyperkähler manifolds that can be obtained from coadjoint orbits by hyperkähler reduction. Jointly with L. Godinho, we showed that these spaces are isomorphic to certain families of parabolic Higgs bundles, when a suitable condition between the parabolic weights and the spectra of the coadjoint orbits is satisfied. In analogy to this construction, we introduce two moduli spaces: the moduli spaces of quasi-parabolic \(SL(2,\mathbb{C})\)-Higgs bundles over \(\mathbb{C}\mathbb{P}^1\) on one hand and the null hyperpolygon spaces on the other, and establish an isomorphism between them. Finally we describe the fixed loci of natural involutions defined on these spaces and relate them to the moduli space of null hyperpolygons in the Minkowski 3-space. This is based on joint works with Leonor Godinho.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Alessia Mandini (Universidade Federal Fluminense, Brasil) - titulo: Kinetic nonholonomic dynamics is neither Hamiltonian nor variational and however... abstract: | It is well-known that kinetic nonholonomic dynamics is neither Hamiltonian nor variational. However, in this talk, after introducing a geometrical description of nonholonomic dynamics, I will present a result which is a little surprising. Namely, for a kinetic nonholonomic system and a given point \(q\) of the configuration space \(Q\), one may define:Many of the new developments in machine learning are connected with gradient-based optimization methods. Recently, these methods have been studied using a variational perspective. This has opened up the possibility of introducing variational and symplectic integration methods using geometric integrators. In particular, in thistle, we introduce variational integrators which allows us to derive different methods for optimization. However, since the systems are explicitly time-dependent, the preservation of the symplecticity property occurs solely on the fibers. Finally, using discrete Lagrange-d'Alembert principle we produce optimization methods whose behavior is similar to the classical Nesterov method reducing the oscillations of typical momentum methods. Joint work with Cédric M. Campos and Alejandro Mahillo.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: David Martín de Diego (Instituto de Ciencias Matemática, España) - titulo: Sobre sistemas mecánicos discretos forzados y la reducción de Routh discreta abstract: |Los sistemas lagrangianos y hamiltonianos con fuerzas aparecen en distintos contextos tales como sistemas con fuerzas de control o sistemas con fuerzas de disipación y fricción. También es usual que los sistemas dinámicos que se obtienen al reducir una simetría de un sistema mecánico sean sistemas forzados. En los casos en los que las fuerzas no puedan ser absorbidas por el lagrangiano o el hamiltoniano como parte de un potencial, se realiza una descripción variacional alternativa del caso sin fuerzas. Como en el caso continuo, cuando se realiza un proceso de reducción de una simetría de un sistema mecánico discreto, es usual que el sistema dinámico reducido presentes términos que se pueden interpretar como fuerzas. Entre otras razones, esto hace que resulte interesante estudiar las características de los sistemas discretos forzados. Un caso particularmente interesante de reducción de simetrías de un sistema mecánico es el que se basa en el uso de sus cantidades conservadas. Cuando se considera un sistema mecánico discreto que admite una aplicación momento que se conserva sobre las trayectorias del sistema, se aplica el proceso conocido como la reducción de Routh discreta que da lugar a un sistema forzado. En esta charla se consideran sistemas mecánicos discretos forzados. A partir de su dinámica definida por un principio de Lagrange d'Alembert discreto que se define modificando convenientemente el principio variacional usual, se estudia la existencia de estructuras simplécticas y su posible conservación por la evolución del sistema. En particular, se considera el proceso de reducción de Routh discreta para una simetría no necesariamente abeliana de un sistema mecánico discreto como un caso particular de un proceso de reducción de simetrías más general. También se analizan la existencia y la conservación de estructuras simplécticas sobre el espacio reducido en el marco de los sistemas forzados teniendo en cuenta las características propias de este caso especial.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Marcela Zuccalli (Universidad Nacional de La Plata, Argentina) - sesion: Teoría de códigos y temas afines zoom: https://us02web.zoom.us/j/81702805205 organizadores: - nombre: Ricardo Podestá mail: podesta@famaf.unc.edu.ar - nombre: Claudio Qureshi mail: cqureshi@gmail.com charlas: - titulo: The conorm code of an AG-code abstract: |Let \(\mathbb{F}_q\) be a finite field with \(q\) elements. For a given trascendental element \(x\) over \(\mathbb{F}_q\), the field of fractions of the ring \(\mathbb{F}_q[x]\) is denoted as \(\mathbb{F}_q(x)\) and it is called a rational function field over \(\mathbb{F}_q\). An (algebraic) function field \(F\) of one variable over \(\mathbb{F}_q\) is a field extension \(F/\mathbb{F}_q(x)\) of finite degree. The \textit{Riemann-Roch space} associated to a divisor \(G\) of \(F\) is the vector space over \(\mathbb{F}_q\) defined as $$\mathcal{L}(G)=\{x\in F\,:\, (x)\geq G\}\cup \{0\},$$ where \((x)\) denotes the principal divisor of \(x\). It turns out that \(\mathcal{L}(G)\) is a finite dimensional vector space over \(\mathbb{F}_q\) for any divisor \(G\) of \(F\). Given disjoint divisors \(D=P_1+\cdots+P_n\) and \(G\) of \(F/\mathbb{F}_q\), where \(P_1,\ldots,P_n\) are different rational places, the \textit{algebraic geometry code} (AG-code for short) associated to \(D\) and \(G\) is defined as $$C_\mathcal{L}^F (D,G) = \{(x(P_1),\ldots, x(P_n))\,:\,x\in \mathcal{L}(G)\}\subseteq (\mathbb{F}_q)^n,$$ where \(x(P_i)\) denotes the residue class of \(x\) modulo \(P_i\) for \(i=1,\ldots,n\).
In this talk the concept of the conorm code, associated to an AG-code will be introduced. We will show some interesting properties of this new code since some well known families of codes such as repetition codes, Hermitian codes and Reed-Solomon codes can be obtained as conorm codes from other more basic codes. We will see that in some particular cases over geometric Galois extensions of function fields, the conorm code and the original code are different representations of the same algebraic geometry code.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: María Chara (Universidad Nacional del Litoral, Argentina) - titulo: Laticce codes abstract: |Lattices are discrete sets of the n-dimensional Euclidean space given by all integer combinations of a set of independent vectors. Lattices have being used in processes of coding for signal transmission over MIMO ( multiple input /multiple output) channels and in cryptographic schemes in the recent area of post-quantum cryptography. In this talk we give a general approach to this topic and present briefly some recent results regarding perfect lattices in different metrics, lattices constructed from codes over finite rings and an extension of the Ring-LWE problem in Cryptography which involves a more general class of algebraic lattices.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Sueli I. R. Costa (Universidade Estadual de Campinas, Brasil) - titulo: The Generalized Covering Radii of Codes abstract: |The generalized covering-radius hierarchy of a linear code is a new property of codes which was motivated by an application to database linear querying, such as private information-retrieval protocols. It characterizes the trade-off between storage amount, latency, and access complexity, in such database systems.
In this work, we introduce and discuss three definitions for the generalized covering radius of a code, highlighting the combinatorial, geometric, and algebraic properties of this concept and prove them to be equivalent. We also discuss a connection between the generalized covering radii and the generalized Hamming weights of codes by showing that the latter is in fact a packing problem with some rank relaxation.
Other contributions to the subject include bounds relating various parameters of codes to the generalized covering radii and an asymptotic upper bound (for particular parameters) which shows that the generalized covering radii improves the naive approach.
Joint work with Dor Elimelech and Moshe Schwartz (Ben-Gurion University/Israel).
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Marcelo Firer (Universidade Estadual de Campinas, Brasil) - titulo: A novel version of group codes abstract: |Let \(G\) be a finite abelian group, with \(G=\prod_{i=1}^n \langle g_{i} \rangle \) where \(|\langle g_{i} \rangle|=m_{i}\). Then every element in \(G\) can be uniquely written as \(\prod_{i=1}^n g_i^{\epsilon_i}\) where \(0\leq \epsilon_i \leq m_{i}-1 \). To determine a measure of the separation between two elements of \(G\) we use the \texttt{Minkowski distance \(l_1\)}, which is given by $$l_{1}\Big(\prod_{i=1}^n g_i^{\epsilon_i}, \prod_{i=1}^n g_i^{\delta_i}\Big) = \sum_{i=1}^n |\epsilon_i-\delta_i|.$$
A grid code \(\mathscr{C}\) is a subset of \(G\), if \(\mathscr{C}\) is subgroup of \(G\), then it is said that \(\mathscr{C}\) is a group code. The elements of \(\mathscr{C}\) are called codewords. The minimum distance \(d\) of a code \(\mathscr{C}\) is defined as usually, that is, as the smallest distance between any two different elements of \(\mathscr{C}\). Let \(\mathscr{C}\) be a code of \(G\) with minimum distance \(d\). Then we say that \(\mathscr{C}\) is a \((n,|\mathscr{C}|,d)\)-code over \(G\) and \((n,|\mathscr{C}|,d)\) are its parameters.
In this talk, we consider such codes, and we prove some classical results on block codes, like Singleton Bound and others.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Ismael Gutiérrez (Universidad del Norte, Colombia) - titulo: Direct sum of Barnes-Wall lattices via totally real number fields abstract: |Let \(\mathbb{K}\) be a number field of degree \(n\), \(\mathcal O_{\mathbb{K}}\) its ring of integers and \(\alpha \in \mathcal O_{\mathbb{K}}\) a totally real and totally positive element. In 1999, Eva Bayer-Fluckiger introduced a twisted embedding \(\sigma_{\alpha}:\mathbb{K} \rightarrow \mathbb{R}^{n}\) such that if \(\mathcal I \subseteq \mathcal O_{\mathbb{K}}\) is a free \(\mathbb{Z}\)-module of rank \(n\), then\(\sigma_{\alpha}(\mathcal I)\) is a lattice in \(\mathbb{R}^{n}\). It was shown that if \(\mathbb{K}\) is a totally real number field, then \(\sigma_{\alpha}(\mathcal I)\) is a full diversity lattice. In this talk we will approach constructions of direct sum of Barnes-Wall lattices \(BW_n\) for \(n=4,8\) and \(16\) via ideals of the ring of the integers \(\mathbb{Z}[\zeta_{2^{r}q} + \zeta_{2^{r}q}^{-1}]\) for \(q = 3, 5\) and \(15\). Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. The minimum product distances of such constructions are also presented here. This is a joint work with Jo\~{a}o E. Strapasson, Agnaldo J. Ferrari and Sueli I. R. Costa and it was partially supported by Fapesp 2013/25977-7, 2014/14449-2 and 2015/17167-0 and CNPq 432735/2016-0 and 429346/2018-2.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Grasiele C. Jorge (Universidade Federal de São Paulo, Brasil) - titulo: AG codes, bases of Riemann-Roch spaces and Weierstrass semigroups abstract: |In this talk we are going to discuss some results on the explicit construction of bases of Riemann-Roch spaces, weierstrass semigroups and the floor of certain divisors. We also will apply these results to get AG codes with good parameters.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Horacio Navarro (Universidad del Valle, Colombia) - titulo: The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs abstract: |We use known characterizations of generalized Paley graphs which are Cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves defined over extension fields. This talk is based on a recent joint work with Ricardo Podestá.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Denis Videla (Universidad Nacional de Córdoba, Argentina) - titulo: A Decoding Algorithm of MDS Array Codes abstract: |This work aims to present a construction of MDS (Maximum Distance Separable) array codes constructed by using superregular matrices, in particular Vandermonde matrices and Cauchy matrices, and the Frobenius companion matrix obtained through a primitive polynomial over \(\mathbb{F}_q[x]\). Array codes are two-dimensional error correction codes whose main characteristic is the ability to correct burst of errors, that is, errors that occur in consecutive bits. MDS codes are codes in which the minimum distance is the maximum possible. This characteristic is important because, in coding theory, the minimum distance is related to the error correction capacity of the code in addition to providing maximum protection against failures of a device for a given amount of redundancy. Based on this construction, a decoding algorithm is presented to correct up to two bursts of errors in MDS array codes with parameters \([m + k, k, m + 1]\) over \(\mathbb{F}_q^b\), for all \(m \geq 4\), where \(b\) refers to the length of the error. In addition, some examples are presented to correct three bursts of errors. This is a joint work with Débora Beatriz Claro Zanitti, São Paulo State University (UNESP).
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Cintya Wink de Oliveira Benedito (Universidade Estadual Paulista, Brasil) - sesion: Análisis no lineal en espacios de Banach zoom: https://us02web.zoom.us/j/81106679177 organizadores: - nombre: Gerardo Botelho mail: botelho@ufu.br - nombre: Daniel Carando mail: dcarando@dm.uba.ar - nombre: Maite Fenández Unzueta mail: maite@cimat.mx charlas: - titulo: Geometría Lipschitz para espacios de operadores abstract: |La teoría no lineal de espacios de Banach ha sido un área muy activa en las últimas décadas, pero su contraparte no conmutativa (es decir, la teoría no lineal de espacios de operadores) apenas está siendo explorada.
En un trabajo previo con Bruno M. Braga hemos mostrado que la noción más obvia de transformación Lipschitz entre espacios de operadores desafortunadamente da lugar a una teoría trivial, en el sentido de que las únicas transformaciones que satisfacen la definición son lineales. En este trabajo, conjunto con Bruno M. Braga y Thomas Sinclair, introducimos una noción más sutil de encaje Lipschitz entre espacios de operadores: estrictamente más débil que la noción lineal, pero suficientemente rígida para imponer restricciones en la estructura lineal de los espacios de operadores. Con este propósito introducimos una noción de espacios libres Lipschitz para espacios de operadores, y probamos algunas de sus propiedades al estilo del trabajo clásico de G. Godefroy y N. Kalton.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Alejandro Chávez Domínguez (University of Oklahoma, Estados Unidos) - titulo: Singular perturbations of symplectic isomorphism of Banach spaces abstract: |Given a real reflexive Banach space \(X\), an onto isomorphism \(\alpha: X \to X^*\) is said to be \emph{symplectic} if \(\alpha^*= - \alpha\) with the canonical identification. We study some properties of strictly singular perturbations of symplectic isomorphism in the direction of the relations obtained by V. Ferenczi and E. Galego (2007) on complex structures of a Banach and its hyperplanes with the elements of square -1 of the algebra \(\mathcal L(X)/S(X)\) quotient of the algebra \(\mathcal L(X)\) of linear and continuous operators on \(X\) by the ideal of strictly singular operators.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Wilson Cuellar Carrera (Universidade de Sâo Paulo, Brasil) - titulo: Normas tensoriales en operator spaces abstract: |En esta charla presentamos el inicio de una teoría sistemática de normas tensoriales en operator spaces y su interacción con ideales de operadores lineales asociados. Basados en la teoría de productos tensoriales en espacios de Banach, proponemos las definiciones naturales correspondientes a este nuevo ámbito y obtenemos algunos resultados análogos. Sin embargo, hay también notables discrepancias con la teoría clásica que requieren nuevos conocimientos, técnicas, ideas o hipótesis. En particular, mostramos varias diferencias sustanciales y muchas preguntas abiertas en la lista de las denominadas normas naturales (en el sentido de Grothendieck).
Trabajo en colaboración con Alejandro Chávez-Domínguez y Daniel Galicer.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Verónica Dimant (Universidad de San Andrés, Argentina) - titulo: Linear dynamics of operators on non-metrizable topological vector spaces abstract: |In this talk we will explore the following notions of chaos for operators on non-metrizable topological vector spaces: mixing, hypercyclicity, cyclicity, \(n\)-supercyclicity, Li-Yorke chaos and Devaney chaos. We will discuss the main differences between these notions for operators on Fr\'echet and on non-metrizable topological vector spaces.
To illustrate some of these differences, we will explore results about the linear dynamics of convolution operators on spaces of entire functions of finitely and infinitely many complex variables. A classical result due to Godefroy and Shapiro states that every nontrivial convolution operator on the Fréchet space \(\mathcal{H}(\mathbb{C}^n)\) of all entire functions of \(n\) complex variables is hypercyclic. In sharp contrast to this result, in a joint work with J. Mujica it was showed that no translation operator on the space \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) (which is a complete non-metrizable locally convex space) of entire functions of infinitely many complex variables is hypercyclic. Recently, in a joint work with B. Caraballo it was showed that no convolution operator on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) is cyclic or \(n\)-supercyclic for any positive integer \(n\). In the opposite direction, it was proved that every nontrivial convolution operator on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) is mixing. Exploring the concept of Li-Yorke chaos on non-metrizable topological vector spaces, it was also proved that nontrivial convolution operators on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) are Li-Yorke chaotic.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Vinícius V. Fávaro (Universidade Federal de Uberlândia, Brasil) - titulo: Constante de proyección de espacios de polinomios abstract: |Un resultado elemental en la teoría de espacios normados asegura que todo espacio de dimensión finita está complementado en cualquier superespacio que lo contenga. Luego, resulta natural querer cuantificar la norma de una determinada proyección sobre un espacio dado.
Una cantidad clásica relacionada con este propósito es la denominada \emph{constante de proyección} del espacio \( X \), definida como la mejor constante \(c>0\) que asegura para \emph{cualquier} superespacio \( Y \supset X\), la existencia de una proyección de \( Y \) sobre \( X \) de norma menor o igual a \( c \).
En esta charla discutiremos la constante de proyección de \(\mathcal P_m(X_n)\) (el espacio de polinomios \(m\)-homogéneos sobre un espacio \(n\)-dimensional \(X_n\)) y la conexión existente con algunos invariantes geométricos relativos a \(X_n\). Nos focalizamos en la dependencia respecto de ambos parámetros \(m\) y \(n\), i.e., el grado de homogeneidad y la dimensión del espacio, respectivamente. También trataremos la constante de proyección para otras clases de polinomios (e.g., polinomios de Dirichlet, polinomios sobre el cubo Booleano, trigonométricos analíticos, polinomios de grado menor o igual a \(m\) y tetraedrales homogéneos, entre otros). Mencionaremos cómo la constante de proyección de espacios de polinomios se conecta con el estudio de incondicionalidad en espacios de polinomios y el radio de Bohr.
Trabajo en conjunto con A. Defant, M. Mansilla, M. Mastyło y S. Muro.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Daniel Galicer (Universidad de Buenos Aires, Argentina) - titulo: Ideals of multilinear operators from a geometric approach abstract: |A recent approach for studying collections of multilinear operators, based on the geometry of tensor products, allows to generalize collections of linear operators to the multilinear setting. Besides defining new collections, this approach allows to characterize them in terms of summing properties, factorizations and duality relations with tensor products. The defined collections to this day include compact, summing, dominated and even factorable operators. These collections of multilinear operators enjoy an ideal behavior, that is, they are closed under compositions that preserves the geometry of the multilinear domain. This ideal property, together with the geometry of tensor products, has inspired the study of the so called \(\Sigma\)-ideals.
This talk is dedicated to present the foundations of \(\Sigma\)-ideals and their classification in terms of tensor norms. Moreover we will explore the \(\Sigma\)-ideal of \((p,q)\)-dominated multilinear operators and its classification in terms of tensor spaces as well as the factorization through \(L_p\) spaces that they admit.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Samuel García Hernández (Universidad Tecnológica de México, México) - titulo: Geometría de los espacios de sucesiones de Marcinkiewicz abstract: |Los espacios de sucesiones de Marcinkiewicz \(m_{\Psi}^0\) y sus duales \((m_\Psi^0)'\) y \(m_\Psi\), son espacios invariantes por reordernamientos, determinados por un símbolo \(\Psi\) (una sucesión no decreciente de números reales positivos). Para símbolos apropiados \(\Psi\), estos espacios devienen en espacios de Lorentz (sus preduales y duales) \(d_*(w,1), d(w,1)\) y \(d^*(w,1)\).
El objetivo de esta charla es entender, para un símbolo general \(\Psi\), la geometría de la bola unidad de \(m_{\Psi}^0,(m_\Psi^0)'\) y \(m_\Psi\) a partir de la caracterización de sus puntos extremales (reales y complejos) y de sus puntos expuestos.
Veremosque los puntos extremales complejos de la bola unidad de \(m_\Psi^0\) están determinados por la geometría de los subespacios finito-dimensionales \(m_\Psi^n\). Mientras que la geometría de la bola unidad de \(m_\Psi\) depende fuertemente del comportamiento asintótico de los valores de \(\Psi\). También caracterizaremos los puntos extremales y expuestos de la bola unidad de \((m^0_\Psi)'\). Como consecuencia, extendemos resultados de Kamińska, Lee y Lewicki (2009) y de Ciesielski y Lewicki (2019). Trabajo en conjunto con Chris Boyd (UCD).
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Silvia Lassalle (Universidad de San Andrés, Argentina) - titulo: Cluster value problems in Banach spaces abstract: |In this talk I will discuss a couple of problems about the Banach algebra \(H^{\infty}(B)\) of bounded holomorphic functions on the ball \(B\) of a Banach space \(X\). First I will talk about the Corona problem, which asks if the evaluations on each point of \(B\) of functions in \(H^{\infty}(B)\) is a dense subset of \(M_{H^{\infty}(B)}\), the nonzero algebra homomorphisms from \(H^{\infty}(B)\) into \(\mathbb{C}\). An easier problem deals with a comparison of the limit behavior of functions in \(H^{\infty}(B)\) towards each point \(x^{**}\) of the closed ball \(\overline{B}^{**}\) of \(X^{**}\) with the elements of \(M_{H^{\infty}(B)}\) that in a sense correspond to \(x^{**}\). I will discuss why the last problem, the cluster value problem, is indeed easier than the Corona problem, and I will talk about some of the essential ideas behind the cluster value theorems known up to now.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Sofía Ortega Castillo (Universidad de Guadalajara, México) - titulo: Spaceability and Residuality in some sets of analytic functions on the open unit disk abstract: |In this talk we will present some results concerning algebraic and topological structure inside the following sets of special functions: the set \(\mathcal{F}\) of all Bloch functions defined on the open unit disk that are unbounded, and the set \(\mathcal{W}^p\) of all Bergman functions defined on the open unit disk that are not Bloch functions. We show that \(\mathcal{F}\) and \(\mathcal{W}^p\) are spaceable and residual, that is, \(\mathcal{F}\cup\{0\}\) (and \(\mathcal{W}^p\cup\{0\}\)) contains a closed infinite dimensional vector space. Residuality of the set \(\mathcal{F}\) (and \(\mathcal{W}^p\)) means that their complements are of first category. We also investigate the set of all holomorphic functions of bounded type defined on a Banach algebra \(E\) into \(E\), which are not Lorch-analytic. We show that this set is spaceable but not residual.
Joint work with M. Lilian Lourenço, IME-USP, São Paulo, Brazil.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Daniela M. Vieira ( Universidade de São Paulo, Brasil) - titulo: Extremos de polinomios - un enfoque probabilístico abstract: |Consideremos un polinomio \(k\)-homogéneo \(P:\mathbb R^n \longrightarrow \mathbb R\). ¿Cuál es la probabilidad de que \(P\) alcance un máximo relativo en algún vértice de la bola-1 (i.e., la bola unidad de la norma \(\Vert \cdot \Vert_1\))? ¿Y en un v\'ertice de la bola-\(\infty\)? Se sabe que si \(k \gt 2\) la probabilidad de alcanzar un máximo relativo en algún vértice de la bola-1 tiende a uno a medida que la dimensión \(n\) crece. Esto es falso para \(k=2\), y es un problema abierto para la bola-\(\infty\).
En esta charla veremos algunas de las herramientas utilizadas para encarar estas cuestiones, y algunas de las dificultades que se presentan.
Veremos también un resultado reciente, obtenido en conjunto con Damián Pinasco y Ezequiel Smucler, para polinomios sobre un simple: si \(k>4\), la probabilidad de que un polinomio \(k\)-homogéneo alcance un máximo relativo en algún vértice del simple \(n\)-dimensional tiende a uno al crecer la dimensión \(n\). Esto requiere un aporte a un viejo problema estadístico: el de las probabilidades ortantes.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Ignacio Zalduendo (Universidad Torcuato Di Tella, Argentina) - sesion: Ecuaciones diferenciales y estructuras geométricas zoom: https://us02web.zoom.us/j/87132938471 organizadores: - nombre: John Alexander Arredondo García mail: alexander.arredondo@konradlorenz.edu.co - nombre: Ronaldo Alves García mail: ragarcia@ufg.br - nombre: Mikhail Malakhaltsev mail: mikarm@uniandes.edu.co - nombre: Jesús Muciño Raymundo mail: muciray@matmor.unam.mx charlas: - titulo: On the averaging theory for computing periodic orbits abstract: |This talk deals with the averaging theory, which is a classical tool allowing us to study periodic solutions of the nonlinear differential systems.
We present some applications of averaging theory. Also, we shall show a system where the averaging method fails to detect periodic orbits.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Martha Alvarez Ramírez (Universidad Autónoma Metropolitana, México) - titulo: Integrable models close to slow-fast Hamiltonian systems abstract: |Slow-fast Hamiltonian systems are characterized by a separation of the phase space into slow and fast parts typically identified by a small (or slow) parameter. This kind of Hamiltonian system is not integrable, in general; even though in one degree of freedom. In this lecture, we study an integrable model associated with a slow-fast Hamiltonian system of two degrees of freedom. Thinking of the slow parameter as a perturbative one, making suitable symmetry assumptions, and using normal form theory, we show that a slow-fast Hamiltonian system in two degrees of freedom is close to an integrable Hamiltonian model. What we gain with this model is the possibility to associate a family of Lagrangian 2-tori which is almost invariant with respect to the original slow-fast Hamiltonian system. As an important application, this family of almost invariant Lagrangian 2-tori can be used to compute approximations to the spectrum of the quantum model associated with the slow-fast Hamiltonian systems.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Misael Avendaño-Camacho (Universidad de Sonora, México) - titulo: The Maximum Principle and solutions to nonlinear elliptic problems abstract: |In this talk, we will use a form of the maximum principle and an iteration method to show existence of solutions to the Dirichlet problem: \[\left\{\begin{array}{l}-\Delta u = \lambda\left(x\right)f\left(u\right)+h\left(x\right)\quad \mbox{in}\quad \Omega,\\u=0 \quad \mbox{on}\quad \partial \Omega,\end{array}\right.\]in bounded and thin unbounded domains. Also, we will show how these methods can be applied to show existence of nontrivial solutions to the Lane-Emden system\[\left\{\begin{array}{l}-\Delta u=v^p,\quad -\Delta v=u^p, \quad \mbox{in}\quad \Omega\\u=v=0, \quad\mbox{on}\quad \partial \Omega. \end{array}\right.\]
This is joint work with Jonatán Torres-Orozco.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Jean C. Cortissoz (Universidad de los Andes, Colombia) - titulo: Singularities of the Focal Set of a Line Congruence abstract: |A line congruence is a \(2\)-dimensional arrangement of lines in \(3\)-space. It is a classical topic of projective differential geometry and has many relations with binary differential equations. In this talk we give a geometric description of the generic singularities of the focal surface of a line congruence. We show that these singularities occur along the ridge and double eigenvalue curves. A basic tool is the support function associated with an eqüiaffine vector field transversal to a surface in \(\mathbb{R}^3\).
This is a joint work with Ronaldo Garcia (UFG, Brazil).
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Marcos Craizer (Pontifícia Universidade Católica do Rio de Janeiro, Brasil) - titulo: Line congruence in singular surface in \(\mathbb R^3\) abstract: |A line congruence in the Euclidean space of dimension \(3\) is a \(2\) parameter family of lines in \(\mathbb R^{3}\).
The first record about line congruences appeared in "Mémoire sur la Théorie des Déblais et des Remblais" (1776,1784) where Gaspard Monge seeks to solve a minimizing cost problem of transporting an amount of land from one place to another, preserving the volume.
After Monge, Ernst Eduard Kummer in ''Allgemeine Theorie der geradlinigen Strahien systeme", was the first to deal exclusively with the general theory of line congruences. Mainly due to optical applications, line congruences started to gain importance, increasing even more with the development of technology. In this lecture, we will deal with line congruence when the parameters vary in a surfaces with singularities. We will build the theory of congruence lines when the parameter space is a Frontal. As application, we will consider line congruence of cuspidal edges. We will show relations between singularities of the congruence lines and geometric properties of initial cuspidal edges.
This a joint work with Tito Medina(ICMC/USP), Igor Chagas Santos(ICMC/USP) and Maria Aparecida S. Ruas(ICMC/USP).
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Débora Lopes da Silva (Universidade Federal de Sergipe, Brasil) - titulo: Frobenius manifolds in the context of \(\mathbb{A}\)-manifolds abstract: |The theory of Frobenius manifolds has important relevance in mathematics since the application to enumerative geometry with the counting of the number of zero genus curves of degree \(d\) passing through \(3d-1\) points in the projective space \(\mathbb{C} P^2\) (M. Kontsevich).
Let \(\mathbb{A}\) be a finite-dimensional commutative associative algebra with unit over \(\mathbb{R}\). A map \(F: \mathbb{A}^n \to \mathbb{A}^n\) is called \(\mathbb{A}\)-differentiable if \(dF\) is \(\mathbb{A}\)-lineal. Let \(\Gamma_\mathbb{A}\) be the pseudogroup of local \(\mathbb{A}\)-diffeomorphisms of \(\mathbb{A}^n\). An \(\mathbb{A}\)-manifold is a manifold endowed with \(\Gamma_\mathbb{A}\)-atlas. An important example of \(\mathbb{A}\)-manifold is the total space of A. Weil's bundle of \(\mathbb{A}\)-closed points.
The theory of \(\mathbb{A}\)-manifolds (A.P. Shirokov, V.V. Vishnevskii, G.I. Kruchkovich, V.V. Shurygin) has strong relations to the foliation theory and to geometry of jet bundles and the theory of natural operations in differential geometry (I. Kolář, J. Slovák, P. W. Michor).
In our talk we will show that Frobenius manifolds in the sense of B. Dubrovin and N.J. Hitchin have the structure of an \(\mathbb{A}\)-manifold, where \(\mathbb{A}\) is a Frobenius algebra. We will introduce Weil coalgebras, the \(G\)-equivariant graduated versions of Weil algebras, and the corresponding bundle structures.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Mikhail Malakhaltsev (Universidad de los Andes, Colombia) y Carlos Segovia (Universidad Nacional Autónoma de México, México) - titulo: Classification of complex polynomials in \(\mathbb{C}^2\) and its applications abstract: |The huge group of polynomial diffeomorphisms of \(\mathbb{C}^2\) acts on the space of polynomials \(\{ f: \mathbb{C}^2 \longrightarrow \mathbb{C} \}\) by coordinate changes. The classification of polynomials under this action is a challenging open problem. We obtain a very concrete result for degree three polynomials. An application to perturbation of singular Hamiltonian foliations on \(\mathbb{C}^2\), having \(\mathbb{C} \backslash \{ k \, \hbox{points}\}\), as generic leafs, is provided.
Joint work with John Alexander Arredondo (Colombia) and Salomón Rebollo (Chile).
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Jesús R. Muciño-Raymundo (Universidad Nacional Autónoma de México, México) - titulo: Umbilic Points at Infinity of Certain Algebraic Surfaces abstract: |In this lecture, we study the global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function \(f\) on the plane. We will prove that every umbilic point at infinity of the projective extension of these direction fields has a Poincar\'e-Hopf index equal to 1/2 and the topological type of a Lemon or a Monstar. As a consequence, we will provide a Poincar\'e-Hopf type formula for the graph of \(f\) pointing out that, if all umbilics are isolated, the sum of all indices of the principal directions at its umbilic points only depends upon the number of real linear factors of the homogeneous part of highest degree of \(f\).
This is a joint work with B. Guilfoyle.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Adriana Ortiz-Rodríguez (Universidad Nacional Autónoma de México, México) - titulo: Perturbations of Hamiltonian foliations abstract: |At the International Congress of Mathematics held in Paris in 1900, D. Hilbert presented a list of 23 problems for the century. After more than one hundred years some of those problems are still unsolved. Among them is the 16th problem. This problem, in its second part, asks for the number and position of limit cycles (isolated periodic solutions) of planar polynomial differential equations. In the 70's, Y. Ilyashenko and V. I. Arnold proposed, in this context, to investigate what happens for systems infinitely close to Hamiltonians. This approach is known as the infinitesimal Hilbert 16th problem. More precisely, let \(F\in\mathbb{R}[x,y]\) a polynomial and \(\omega\) a polynomial 1-form. We consider the perturbation \(dF+\epsilon\omega=0\) of the Hamiltonian foliation \(dF=0\), where \(\epsilon\) is a small parameter. Let \(\gamma(z)\subset F^{-1}(z)\) a continuous family of regular periodic orbits of \(dF=0\). The displacement map with respect to this family \(\gamma(z)\) and with respect to the foliation \(dF+\epsilon\omega=0\) is an analytic function \(\Delta(z,\epsilon)=\epsilon^\mu M_\mu(z)+\cdots\). The first function \(M_\mu\) which does not vanish identically keeps information about limit cycles born from this perturbation. By passing to the complexification of \(F\), we can consider the orbit under monodromy of \(\gamma(z)\), which is a normal subgroup \(\mathcal{O}\) of the first homotopy group of the regular fiber \(F^{-1}(z)\). To understand the orbit under monodromy one must focus on the critical values of the function \(F\). To each critical value corresponds a vanishing cycle; the monodromy related to such cycle detects the traces of change in the topology due to the presence of the singular fiber. In non generic cases, not every cycle can be reached by the orbit under monodromy of the family of regular periodic orbits \(\gamma(z)\). Thus, to detect the cycles that are not reached one considers the quotient \(\frac{\mathcal{O}}{[\mathcal{O},\pi_1(F^{-1}(z),p_0)]}\) and defines a constant \(\kappa\), called orbit depth. It turns out that the function \(M_\mu\) is an iterated integral of length at most \(\kappa\). We stress that \(\kappa\) by its definition depends only on the topology of the regular fiber of \(F\), and on the orbit \(\mathcal{O}\) of \(\gamma(z)\), while the function \(M_\mu\) depends also on the perturbation given by \(\omega\). We will talk about this result and discuss some conjectures around this bound for non-generic polynomials.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Jessie Pontigo Herrera (Universidad Nacional Autónoma de México, México) - titulo: Solutions of algebraic linear ordinary differential equations abstract: |A classical result of F. Klein states that, given a finite primitive group \(G\subseteq SL_2(\mathbb{C})\), there exists a hypergeometric equation such that any second order LODE whose differential Galois group is isomorphic to \(G\) is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this paper, we generalize this result. We show that, given a finite primitive group \(G\subseteq SL_n(\mathbb{C})\), there exist a positive integer \(d=d(G)\) and a standard equation such that any LODE whose differential Galois group is isomorphic to \(G\) is gauge equivalent, over a field extension \(F\) of degree \(d\), to an equation projectively equivalent to the pullback by a map in \(F\) of this standard equation. For \(n=3\), these standard equations can be chosen to be hypergeometric.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Camilo Sanabria (Universidad de Los Andes, Colombia) - sesion: Quantum symmetries zoom: https://us02web.zoom.us/j/85136861142 organizadores: - nombre: César Galindo mail: cn.galindo1116@uniandes.edu.co - nombre: Julia Plavnik mail: jplavnik@iu.edu - nombre: Leandro Vendramin mail: lvendramin@dm.uba.ar charlas: - titulo: 'On finite GK-dimensional Nichols algebras of diagonal type: rank 3 and Cartan type' abstract: |It was conjectured by Andruskiewitsch, Angiono and Heckenberger that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system; they did in fact prove this for braidings of affine type or when the rank is two. We shall review some tools developed with the intention of proving this conjecture positively, in a work of Angiono and the author, and exhibit the proof for the rank 3 case, as well as for braidings of Cartan type.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Agustín García Iglesias (Universidad Nacional de Córdoba, Argentina) - titulo: A diagrammatic Carlsson-Mellit algebra abstract: |The \(A_{q,t}\) algebra was introduced by Carlsson and Mellit in their proof of the celebrated shuffle theorem, which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This algebra arises as an extension of two copies of the affine Hecke algebra by certain raising and lowering operators. Carlsson and Mellit constructed an action via plethystic operators on the space of symmetric functions which was then realized geometrically on parabolic flag Hilbert schemes by them and Gorsky. The original algebraic construction was then extended to an infinite family of actions by Mellit and shown to contain the generators of elliptic Hall algebra. However, despite the various formulations of \(A_{q,t}\), performing computations within it is complicated and non-intuitive.
In this talk I will discuss joint work with Matt Hogancamp where we construct a new topological formulation of \(A_{q,t}\) (at t=-1) and its representation as certain braid diagrams on an annulus. In this setting many of the complicated algebraic relations of \(A_{q,t}\) and applications to symmetric functions are trivial consequences of the skein relation imposed on the pictures. In particular, many difficult computations become simple diagrammatic manipulations in this new framework. This purely diagrammatic formulation allows us to lift the operators as certain functors, thus providing a categorification of the \(A_{q,t}\) action on the derived trace of the Soergel category.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Nicolle Gonzalez (The University of California Los Angeles, Estados Unidos) - titulo: Quantum Symmetries in Conformal Field Theory abstract: |Quantum groups and Nichols algebras appear as symmetries in conformal field theories, as so-called non-local screening operators. I will start by giving a motivation of conformal field theory and explain how this leads from the analysis side to a modular tensor category, quite often one that is well known from algebra. I then explain my recent work that non-local screening operators generate Nichols algebras in this category, and also some recent work with T. Creutzig and M. Rupert that conclude a braided category equivalence between a certain physical model and the representation category of the smallest quantum group \(u_q(\mathfrak{sl}_2),\;q^4=1\).
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Simon Lentner (Universität Hamburg, Alemania) - titulo: Slack Hopf monads abstract: |Hopf monads are generalizations of Hopf algebras that bring to bear the rich theory of monads from category theory. Even though this generalization overarches many Hopf-like structures, quasi-Hopf algebras seem to be out of its reach. This talk reports on advances in the study of slack Hopf monads, including how these generalize quasi-Hopf algebras. (Joint work with A. Bruguieres and M. Haim).
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Ignacio Lopez Franco (Universidad de la República, Uruguay) - titulo: Interpolations of monoidal categories by invariant theory abstract: |In this talk, I will present a recent construction that enables one to interpolate symmetric monoidal categories by interpolating algebraic structures and their automorphism groups. I will explain how one can recover the constructions of Deligne for categories such as Rep(S_t), Rep(O_t) and Rep(Sp_t), the constructions of Knop for wreath products with S_t and GL_t(O_r), where O_r is a finite quotient of a discrete valuation ring, and also the TQFT categories recently constructed from a rational function by Khovanov, Ostrik, and Kononov.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Ehud Meir (University of Aberdeen, Escocia) - titulo: Examples of module categories, the non-semisimple case abstract: |In this talk we present classical examples of exact indecomposable module categories over a finite non-semisimple rigid tensor category, as well as recently constructed new examples.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Adriana Mejía Castaño (Universidad del Norte, Colombia) - titulo: Algebras in group-theoretical fusion categories abstract: |The categorical version of a module over a ring is the notion of a module category over a fusion category. We will first discuss what it means for a module category to be represented by an algebra and introduce the notion of Morita equivalence of algebras in fusion categories. Sonia Natale and Victor Ostrik described algebras representing the Morita equivalence classes in pointed fusion categories. We will explain how this result can be generalized to group-theoretical fusion categories, based on joint work with Monique Müller, Julia Plavnik, Ana Ros Camacho, Angela Tabiri, and Chelsea Walton.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Yiby Morales (Universidad de los Andes, Colombia) - titulo: Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge abstract: |I will discuss recent work with T. Gannon in which we provide a ribbon tensor equivalence between the representation category of small quantum SL(2) at q=exp(pi*i/p) and the representation category of the triplet VOA at a corresponding central charge 1-6(p-1)^2/p. Such an equivalence was conjectured in work of Gainutdinov, Semikhatov, Tipunin, and Feigin from 2006.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Cris Negron (University of North Carolina, Estados Unidos) - titulo: Frobenius-Schur indicators for some families of quadratic fusion categories abstract: |Quadratic categories are fusion categories with a unique non-trivial orbit from the tensor product action of the group of invertible objects. Familiar examples are the near-groups (with one non-invertible object) and the Haagerup-Izumi cate- gories (with one non-invertible object for each invertible object). Frobenius-Schur indicators are an important invariant of fusion categories generalized from the theory of finite group representations. These indicators may be computed for objects in a fusion category C using the modular data of the Drinfel’d center Z(C) of the fusion category, which is itself a modular tensor category. Recently, Izumi and Grossman provided new (conjectured infinite) families of modular data that include the modular data of Drinfel’d centers for the known quadratic fusion categories. We use this information to compute the FS indicators; moreover, we consider the relationship between the FS indicators of objects in a fusion category C and FS indicators of objects in that category’s Drinfel’d center Z(C).
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Henry Tucker (University of California at Riverside, Estados Unidos) - sesion: Sistemas Dinámicos y Teoría Ergódica zoom: https://us02web.zoom.us/j/89298796225 organizadores: - nombre: Jairo Bochi mail: jairo.bochi@mat.uc.cl - nombre: Katrin Gelfert mail: gelfert@im.ufrj.br - nombre: Rafael Potrie mail: rpotrie@cmat.edu.uy charlas: - titulo: Multiplicative actions and applications abstract: |In this talk, I will discuss recurrence problems for actions of the multiplicative semigroup of integers. Answers to these problems have consequences in number theory and combinatorics, such as understanding whether Pythagorean trios are partition regular. I will present in general terms the questions, strategies from dynamics to address them and mention some recent results we obtained. This is joint work with Anh Le, Joel Moreira, and Wenbo Sun.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Sebastián Donoso (Universidad de Chile, Chile) - titulo: Actions of abelian-by-cyclic groups on surfaces abstract: |I'll discuss some results and open questions about global rigidity of actions of certain solvable groups (abelian by cyclic) on two dimensional manifolds. Joint work with Jinxin Xue.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Sebastián Hurtado-Salazar (University of Chicago, Estados Unidos) - titulo: Polynomial decay of correlations of geodesic flows on some nonpositively curved surfaces abstract: |We consider a class of nonpositively curved surfaces and show that their geodesic flows have polynomial decay of correlations. This is a joint work with Carlos Matheus and Ian Melbourne.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Yuri Lima (Universidade Federal do Ceará, Brasil) - titulo: Continuity of center Lyapunov exponents. abstract: |The continuity of Lyapunov exponents has been extensively studied in the context of linear cocycles. However, there are few theorems that provide information for the case of diffeomorphisms. In this talk, we will review some of the known results and explain the main difficulties that appear when trying to adapt the usual techniques to the study of center Lyapunov exponents of partially hyperbolic diffeomorphisms.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Karina Marín (Universidade Federal de Minas Gerais, Brasil) - titulo: On tilings, amenable equivalence relations and foliated spaces abstract: |I will describe a family of foliated spaces constructed from tllings on Lie groups. They provide a negative answer to the following question by G.Hector: are leaves of a compact foliated space always quasi-isometric to Cayley graphs? Their construction was motivated by a profound conjecture of Giordano, Putnam and Skau on the classification, up to orbit equivalence, of actions of countable amenable groups on the Cantor set. I will briefly explain how these examples relate to the GPS conjecture. This is joint work with Fernando Alcalde Cuesta and Álvaro Lozano Rojo.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Matilde Martínez (Universidad de la República, Uruguay) - titulo: Lyapunov exponents of hyperbolic and partially hyperbolic diffeomorphisms abstract: |If \(f\) is a diffeomorphism on a compact \(d\)-dimensional manifold \(M\) preserving the Lebesgue measure \(\mu\), then Oseledets Theorem tells us that almost every point has \(d\) Lyapunov exponents (possibly repeated): $$\lambda_1(f,x)\leq\lambda_2(f,x)\leq\dots\leq\lambda_d(f,x).$$ If furthermore \(\mu\) is ergodic, then the Lyapunov exponents are independent of the point \(x\) (a.e.). We are interested in understanding the map $$f\in Diff_{\mu}^r(M)\ \mapsto\ (\lambda_1(f),\lambda_2(f),\dots,\lambda_d(f)),\ r\geq 1.$$ In general this map may be very complicated. However, if we restrict our attention to the set of Anosov or partially hyperbolic diffeomorphisms, then we can understand this map better. I will present various results related to the regularity, rigidity and flexibility of the Lyapunov exponents in this setting.
Some of the results presented are joint with C. Vasquez, F. Valenzuela, J. Yang and P. Carrasco.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Radu Saghin (Pontificia Universidad Católica de Valparaiso, Chile) - titulo: Zero Entropy area preserving homeomorphisms on surfaces abstract: |We review some recent results describing the behaviour of homeomorphisms of surfaces with zero topological entropy. Using mostly techniques from Brouwer theory, we show that the dynamics of such maps in the sphere is very restricted and in many ways similar to that of an integrable flow. We also show that many of these restrictions are still valid for \(2\)-torus homeomorphisms.
start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Fabio Tal (Universidade de São Paulo, Brasil) - titulo: Conjugacy classes of big mapping class groups abstract: |A surface \(S\) is big if its fundamental group is not finitely generated. To each big surface one can associate its mapping class group, \(\mathrm{Map}(S)\), which is \(\mathrm{Homeo}(S)\) mod isotopy. This is a Polish group for the compact-open topology. In this talk we study the action of \(\mathrm{Map}(S)\) on itself by conjugacy and characterize when this action has a dense or co-meager orbit. This is a joint work with Jesus Hernández Hernández, Michael Hrusak, Israel Morales, Anja Randecker and Manuel Sedano (arxiv.org/abs/2105.11282v2).
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Ferrán Valdez (Universidad Nacional Autónoma de México, México) - sesion: Funciones especiales, polinomios ortogonales y teoría de aproximación zoom: https://us02web.zoom.us/j/87219359340 organizadores: - nombre: Manuel Domínguez de la Iglesia mail: mdi29@im.unam.mx - nombre: Pablo Manuel Román mail: roman@famaf.unc.edu.ar - nombre: Actividad relacionada con la Red Iberoamericana de Investigadores en Polinomios Ortogonales, Ecuaciones funcionales y Aplicaciones RIPOEFA charlas: - titulo: Asymptotic analysis of matrix orthogonal polynomials abstract: |In this talk we consider matrix orthogonal polynomials (MVOPs) on a finite interval of the real line, with Jacobi-type weights. We are particularly interested in the asymptotic behavior as the degree of the polynomials tends to infinity, that we study with the method of steepest descent applied to the corresponding Riemann-Hilbert problem, as described by Grünbaum, de la Iglesia and Martínez-Finkelshtein. We include examples motivated by group theory.
This is joint work with Arno Kuijlaars (KU Leuven, Belgium) and Pablo Román (Universidad de Córdoba, Argentina).
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Alfredo Deaño (Universidad Carlos III de Madrid, España) - titulo: Sobolev spaces on graded groups abstract: |Fractional Calculus is one of the areas where Special Functions naturally arise. We will characterize fractional Sobolev spaces of potential type on graded groups. The approach is based on the Littlewood-Paley g-function. This is a joint work with Pablo de Nápoli.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Rocío Diaz Martín (Universidad Nacional de Córdoba, Argentina) - titulo: Elliptic Kac-Sylvester matrix from difference Lamé equation abstract: |Through a finite-dimensional reduction of the difference Lamé equation, an elliptic analog of the Kac-Sylvester tridiagonal matrix is found. We solve the corresponding finite discrete Lamé equation by constructing an orthogonal basis of eigenvectors for this novel elliptic Kac-Sylvester matrix. (Based on work in collaboration with Tamás Görbe.)
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Jan Felipe van Diejen (Universidad de Talca, Chile) - titulo: Sobolev Type orthogonal polynomials on the cone of revolution abstract: |We present the Sobolev Type orthogonal polynomials of several variables defined on the cone and on the surface of the cone as a particular case of the polynomials on a quadratic surface of revolution. Using the results from [1], [2] and [3], we show that this kind of polynomials satisfy some properties.
[1] L. Fernández, T. E Pérez, M. Piñar and Y. Xu. Krall-type orthogonal polynomials in several variables. Comp. Appl. Math. Vol 81. Pags 1519-1524. (2001).
[2] Y. Xu, Sobolev orthogonal polynomials defined via gradient on the unit ball, J. Approx. Theory. 152 (2008), 52--65. (2006).
[3] Y. Xu, Fourier series in orthogonal polynomials on a cone of revolution, arXiv:1905.07587 2019
In this contribution, we present some algebraic and analytic properties related to spectral transformations of orthogonality matrix measures on the unit circle, that constitute generalizations from well-known results on the scalar case. We focus our attention on the so-called Christoffel, Geronimus and Uvarov transformations, and deal with connection formulas, factorizations of block Hessenberg and CVM matrices, and relative asymptotics for the associated orthogonal matrix polynomials. This is a joint work with Edinson Fuentes.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Luis E. Garza Gaona (Universidad de Colima, México) - titulo: Toda lattice, special functions and their matrix analogues abstract: |The classical Toda lattice is a model for a one-dimensional crystal. After a transformation in Flaschka coordinates there exists a Lax pair, for which the operator acts as a three-term recurrence operator. This gives a link to orthogonal polynomials, special functions and Lie algebra representations. In the case of orthogonal polynomials, the time dependence in the Toda lattice corresponds to deformation of the orthogonality measure by an exponential. The nonabelian Toda lattice is a generalisation of the Toda lattice for which matrix valued orthogonal polynomials play a similar role. We discuss matrix polynomials, and we discuss an explicit example of such a nonabelian Toda lattice.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Erik Koelink (Radboud Universiteit, Países Bajos) - titulo: Multiple orthogonal polynomials with respect to hypergeometric functions abstract: |In this talk I will discuss on two new sets of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Confluent and Gauss' hypergeometric functions. The former on unbounded support and the latter with bounded support on the real line. It was recently shown in [3] that the latter are indeed random walk polynomials. In both cases, this type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total-positivity problems in enumerative combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indices lie on the step-line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property.
References:
[1] H. Lima and A. Loureiro, Multiple orthogonal polynomials with respect to Gauss’ hypergeometric function, to appear in Stud. Appl. Math. arXiv:2001.06820.
[2] H. Lima and A. Loureiro, Multiple orthogonal polynomials associated with confluent hypergeometric functions, J. Approx. Theory, 260 (2020), pp 105484.
[3] A. Branquinho, J. E. Fernández-Díaz, A. Foulquié-Moreno and M. Mañas, Hypergeometric Multiple Orthogonal Polynomials and Random Walks, (2021) arXiv:2107.00770v2
We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as Szegö polynomials. These results have connections to classical results from algebraic and projective geometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numerical ranges of a class of matrices; and Blaschke products and disk functions.
This is a joint work with Markus Hunziker, Taylor Poe, and Brian Simanek, all at Baylor University.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Andrei Martínez Finkelshtein (Baylor University, Estados Unidos, y Universidad de Almería, España) - sesion: Teoría algebraica de formas cuadráticas sobre anillos zoom: https://us02web.zoom.us/j/83704861521 organizadores: - nombre: Hugo L. Mariano mail: hugomar@ime.usp.br - nombre: Alejandro Petrovich mail: apetrov@dm.uba.ar charlas: - titulo: Rings of Formal Power Series and Symmetric Real Semigroups abstract: |The aim of this talk is to present, firstly, a number of results on the relationship between the ring \(A = F[[G]]\) of formal power series with coefficients in a formally real (i.e., orderable) field \(F\) and exponents in the positive cone of a totally ordered abelian group \(G\), and the real semigroup \(G_{\!A}\) associated to the ring \(A\). One of the main results shows that the real semigroup \(G_{\!A}\) is a fan in the category of real semigroups if and only if the preorder \(\Sigma F^{2}\) of \(F\) is a fan in the sense of fields. On the other hand, in the general case (i.e., for an arbitrary formally real field of coefficients), the real semigroup \(G_{\!A}\) satisfies certain fundamental conditions formulated in terms of the specialization partial order \(\leadsto\) defined in the real spectrum \({\mathrm{Sper}}(A)\) of the ring \(A\). These properties led us to introduce a new class of real semigroups which we baptized symmetric real semigroups.
Next, we investigate the theory of symmetric real semigroups and prove several results on their structure, leading to the following:
I) Every finite symmetric real semigroup is realizable by a ring of formal power series.
II) There exists an infinite symmetric real semigroup (in fact, a fan) which is not realizable by any ring of formal power series.
If \(G\) is a real semigroup (RS), write \(G^\times = \{ x \in G : x^2 = 1\}\) for the group of units in \(G\) and
\(Id(G) = \{ e \in G : e^2 = e \}\) for the distributive lattice of idempotents in \(G\)
Our purpose here is fourfold: firstly to give, employing the languages of special groups (SG) and real semigroups, new, oftentimes conceptually different and clearer, proofs of the characterization of RSs whose space of characters is Boolean in the natural Harrison (or spectral) topology, originally appearing in section 7.6 and 8.9 of Marshall (1996), therein treated as zero-dimensional abstract real spectra and here called named Boolean Real Semigroups. Secondly, to give a natural {\em Horn-geometric} axiomatization of Boolean RSs (in the language of RSs) and establish the closure of this class by certain important constructions: Boolean powers, arbitrary filtered colimits, products, reduced products and RS-sums and by surjective RS-morphisms (in particular, quotients). Thirdly, to characterize morphisms between Boolean RSs: if \(G\), \(H\) are Boolean RSs, there is a natural bijective correspondence between \(Mor_{RS}(G, H)\) and the set of pair of morphisms, \(\langle f, h \rangle\), where \(f\) is an RSG-morphism from \(G^\times\) to \(H^\times\) and \(h\) is a lattice morphism from \(Id(G)\) to \(Id(H)\), satisfying a certain compatibility condition. Fourthly, to give a characterization of quotients of Boolean RSs. Hence, the present work considerably extends the one by which it was motivated, namely the references in Marshall (1996) mentioned above.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Francisco Miraglia (Universidade de São Paulo, Brasil), joint with Hugo R. O. Ribeiro (USP) - titulo: Ranges of functors and geometric classes abstract: |The representation problem for special groups, asking whether every reduced special group is isomorphic to the special group of a field, is an outstanding open problem in the realm of quadratic forms. The same is true for the corresponding problem about real semigroups, as well as Efrat's question which \(\kappa\)-structures arise as the Milnor \(K\)-theory of a field, are outstanding open problems in the realm of quadratic forms. All of these problems allow variants where one replaces isomorphism by elementary equivalence, e.g. by the question whether every reduced special group is elementarily equivalent to one coming from a field.
Motivated by these problems, we study the general question when the essential image of a functor can be axiomatized by \(\kappa\)-geometric sequents -a certain fragment of formulas of infinitary first order logic.
We observe that one can study this question via topos theory: Under mild hypotheses, functors between accessible categories (such as categories of models of first order theories) can be assumed to be induced by a \(\kappa\)-geometric morphism between classifying \(\kappa\)-toposes, notions first stdied by Espíndola. We show how this morphism can be factorized into a surjection, followed by a dense inclusion, followed by a closed inclusion, and explain what that means in terms of the involved theories. We arrive at very concrete axiomatizability criteria using this factorization.
All results and involved notions will be explained and backed up with examples.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Peter Arndt (Universität Düsseldorf, Alemania) - titulo: Von Neumann Hull for Real Semigroups abstract: |The theory of Real Semigroups (RS) was created by M. Dickmann and A. Petrovich in the 2000's as a first order theory of real spectra of rings. It extends the concept of Reduced Special Group by using representation (or transversal representation) of dimension 2 form as primitive concept.
In this talk we will build the von Neumann Hull of a RS and describe its mains algebraic and categorical properties. As an application, we will prove a version of Marshall conjecture for real semigroups associated with (semi-real) rings.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Hugo Rafael de Oliveira Ribeiro (Universidade de São Paulo, Brasil), joint with Hugo L. Mariano (USP) - titulo: K-theories, Graded Rings and Quadratic Forms abstract: |The uses of K-theoretic (and Boolean) methods in abstract theories of quadratic forms has been proved a very successful method, see for instance, these two papers of M. Dickmann and F. Miraglia: (1998) where they give an affirmative answer to Marshall's Conjecture, and (2003), where they give an affirmative answer to Lam's Conjecture.
These two central papers makes us take a deeper look at the theory of Special Groups by itself. This is not mere exercise in abstraction: from Marshall's and Lam's Conjecture many questions arise in the abstract and concrete context of quadratic forms.
There are some generalizations of Milnor's K-theory. In the quadratic forms context, the most significant one is the Dickmann-Miraglia's K-theory of Special Groups. It is a main tool in the proof of Marshall's and Lam's Conjecture.
In Marshall's paper (2006), he propose a new abstract theory of quadratic forms based on what he called a ``real reduced multiring'' and ``real reduced hyperfield''. This new theory has the advantage of brings new analogies with commutative algebra, and we developed and expand the details in Roberto, Ribeiro, Mariano (2020). It is even possible rewrite the axioms of special groups in a sort of "geometric manner" via hyperfields (Roberto, Ribeiro, Mariano (2021)).
Now, we will give another step in the ``marriage of multi structures and quadratic forms'' developing an appropriate K-theory for hyperfields. This new category generalizes simutaneously both Milnor's reduced and non-reduced K-theories and Dickmann-Miraglia's K-theory for special groups.
With these three K-theories on hands, it is desirable (or, at least, suggestive) the rise of an abstract enviroment that encapsule all them, and of course, provide an axiomatic approach to guide new extensions of the concept of K-theory in the context of the algebraic and abstract theories of quadratic forms. The inductive graded rings, introduced by M. Dickmann and F. Miraglia (2000) fits this purpose, and we finish this work showing that the K-theory of pre-special hyperfields is some kind of free inductive graded ring.
start: 2021-09-14T16:45 end: 2021-09-14T16:45 speaker: Kaique Matias de Andrade Roberto (Universidade de São Paulo, Brasil), joint with Hugo L. Mariano (USP) - titulo: Logical and categorial aspects of abstract quadratic forms theories abstract: |The relationship between Galois groups of fields with orderings and quadratic forms, established by the works of Artin-Schreier (1920's) and Witt (late 1930's) are reinforced by a seminal paper of John Milnor (1971) through the definition of a (mod 2) k-theory graded ring that "interpolates" the graded Witt ring and the cohomology ring of fields: the three graded rings constructions determine functors from the category of fields where 2 is invertible that, almost tree decades later, are proved to be naturally isomorphic by the work of Voevodsky with co-authors.
Since the 1980's, have appeared many abstract approaches to the algebraic theory of quadratic forms over fields that are essentially equivalent (or dually equivalent): between them we emphasize the (first-order) theory of special groups developed by Dickmann-Miraglia. The notions of (graded) Witt rings and k-theory are extended to the category of Special groups with remarkable pay-offs on questions on quadratic forms over fields.
In this talk we extended to (well-behaved) Special Groups the work of J. Minác and Spira that describes a (pro-2)-group of a field extension that encodes the quadratic form theory of a given field \(F\): Adem, Karagueuzian, J. Minác (1999) it is shown that its associated cohomology ring is contains a copy of the cohomology ring of the field \(F\). Our construction, a contravariant functor \(G \in SG \ \mapsto\ Gal(G)\in Pro-2-groups\), encodes the space of orders of the special group \(G\) and provides a criteria to detect when \(G\) is formally real or not. This motivate us to consider tree categories which are endowed with a underlying functor into the category of "pointed" groups of exponent 2: the category of pre-special groups, a category formed by certain pointed graded rings and a category given by some pairs of profinite 2-groups and a clopen subgroup of index at most 2 and with arrows the continuous homomorphisms compatible with this additional data. We establish precise (and canonical) functorial relationship between them and explore some of its model-theoretical aspects.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Hugo Luiz Mariano (Universidade de São Paulo, Brasil), joint with Kaique M. A. Roberto (USP) - sesion: Lógica matemática zoom: https://us02web.zoom.us/j/85681787449 organizadores: - nombre: Manuela Busaniche mail: mbusaniche@santafe-conicet.gov.ar - nombre: Marcelo Esteban Coniglio mail: coniglio@cle.unicamp.br charlas: - titulo: Filter pairs and natural extensions of logics abstract: |We adjust the notion of finitary filter pair ( Finitary Filter Pairs and Propositional Logics, South American Journal of Logic 4(2)), which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality \(\kappa\), {where \(\kappa\) is a regular cardinal}. The corresponding new notion is called \(\kappa\)-filter pair. We show that any \(\kappa\)-filter pair gives rise to a logic of cardinality \(\kappa\) and that every logic of cardinality \(\kappa\) comes from a \(\kappa\)-filter pair. We use filter pairs to construct natural extensions for a given logic and work out the relationships between this construction and several others proposed in the literature ( A note on natural extensions in abstract algebraic logic, Studia Logica, 103(4); Constructing natural extensions of propositional logics, Studia Logica 104(6)). Conversely, we describe the class of filter pairs giving rise to a fixed logic in terms of the natural extensions of that logic.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Hugo Luiz Mariano (Universidade de São Paulo, Brasil), joint with P. Arndt (University of Düsseldorf, Germany) and D. C. Pinto (Federal University of Bahia, Brazil) - titulo: A topological duality for monotone expansions of semilattices abstract: |The aim of this work is to present a topological duality for the variety of monotone semilattices. To do this, combining the duality developed by Celani and González for semilattices together with the approach given by Celani and the author for the study of monotone distributive semilattices, we introduce a category of multirelational spaces called \(m\)S-spaces and we prove that this is dually equivalent to the category of monotone semilattices with homomorphisms. Moreover, as an application of this duality we provide a characterization of congruences of (monotone) semilattices by means of lower Vietoris type topologies. It is worth mentioning that a key tool for developing this duality is the topological description of canonical extension of semilattices that we provide. This is a joint work with Ismael Calomino (CIC and Universidad Nacional del Centro) and William J. Zuluaga Botero (Universidad Nacional del Centro).
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: María Paula Menchón (Universidad Nacional del Centro de la Provincia de Buenos Aires, Argentina) - titulo: Embeddability between orderings and GCH abstract: |We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete cardinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure hypothesis for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings.
start: 2021-09-17T15:45 end: 2021-09-15T16:30 speaker: Rodrigo de Alvarenga Freire (Universidade de Brasília, Brasil) - titulo: Hilbert's tenth problem in rings of meromorphic functions abstract: |We will give a short survey on definability of the integers in rings of meromorphic functions, together with some ideas on the ingredients in the proofs and the reasons why the case of complex entire functions, as a ring, remain an open problem. We will then present two new evidences for the latter to be undecidable, one obtained with T. Pheidas, and the other one with D. Chompitaki, N. García-Fritz, H. Pasten and T. Pheidas.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Xavier Vidaux (Universidad de Concepción, Chile) - titulo: Existentially closed measure preserving actions of the free group. abstract: |In joint work with Ward Henson and Tomás Ibarlucía, we prove that the theory of atomless probability algebras expanded with a measure preserving action of the free group \(F_k\) has a model companion for any \(k \geq 1\) and the corresponding theory is stable. We build explicitly examples of existentially closed models. Our approach also applies to a reasonably large class of \(\aleph_0\)-categorical superstable theories.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Alexander Berenstein (Universidad de los Andes, Colombia) - titulo: Completitud estándar fuerte (fuerte finita) para lógicas de S5-modales de Łukasiewicz abstract: |En esta charla mostraremos una versión algebraica de teoremas de completitud estándar fuerte finita para lógicas de S5-modales de Łukasiewicz y algunas de sus extensiones axiomáticas de interés. Para esto mostraremos que la variedad de la MV-álgebras monádicas están generadas como cuasivariedad por cierta álgebra funcional estándar. Lo mismo se hará para ciertas extensiones axiomáticas de interés. Por último agregaremos una regla infinitaria y mostraremos la completitud estándar fuerte para esa lógica usando nuevamente técnicas algebraicas.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: José Patricio Díaz Varela (Universidad Nacional del Sur, Argentina) - titulo: Distributivity, Modularity, and Natural Deduction abstract: |On the one hand, from a logical point of view, we will pay attention to discussions around the notion of distributivity in the nineteenth century world of the algebra of logic. In particular, we will consider some passages from the writings of Ernst Schršoder and Charles Peirce. On the other hand, from an algebraic point of view, we will trace the origins of the notion of modularity in lattice theory. This we will do examining some writings of Richard Dedekind. Moreover, we will includesome remarks on the different ways to consider the notions of modularity and distributivity in the context of a join semilattice. In the end, paying special attention to the disjunction elimination rule in natural deduction, we will include considerations on those notions in the context of the conjunction-disjunction-negation fragment of propositional intuitionistic logic.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Rodolfo C. Ertola-Biraben (Universidad de Campinas, Brasil) - titulo: Interpolation and Beth definability for conic idempotent Full Lambek calculus abstract: |The Full Lambek calculus forms the basic system for substructural logics, examples of which include relevance, linear, many-valued, intuitionistic and classical logic. We focus on substructural logics that satisfy contraction, mingle and the rule of conicity.
The algebraic semantics of substructural logics are residuated lattices and they have an independent history in the context of order algebra; they were first introduced by Ward and Dilworth as tools in the study of ideal lattices of rings. Residuated lattices have a monoid and a lattice reduct, as well as division-like operations; examples include Boolean algebras, lattice-ordered groups and relation algebras. They are also connected to mathematical linguistics and computer science (for example pointer management and memory allocation). We focus on idempotent conic residuated lattices, which are related to algebraic models of relevance logic, and which extend the class of idempotent linear ones. After establishing a decomposition result for this class, we show that it has the strong amalgamation property, and extend the result to the variety generated by this class; this implies that the corresponding logic has the interpolation property and the Beth definability property.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Nick Galatos (University of Denver, Estados Unidos), joint with Wesley Fussner - titulo: Lindström theorems in graded model theory abstract: |Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this talk I will present some first steps towards an abstract formulation of this model theory. I will give a general notion of abstract logic based on many-valued models and discuss six Lindstr\"{o}m-style characterizations of maximality of first-order logics in terms of metalogical properties such as compactness, abstract completeness, the L\"{o}wenheim-Skolem property, the Tarski union property, and the Robinson property, among others. The results have been obtained in a joint work with Guillermo Badia.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Carles Noguera i Clofent (Czech Academy of Sciences, República Checa) - titulo: Locally pseudocomplemented abelian \(\ell\)-groups abstract: |It is a known result of Glass and Pierce that the variety of abelian lattice ordered groups (\(\ell\)-groups) does not to have a model companion. Call and abelian \(\ell\)-group\ \emph{locally pseudocomplemented} if any bounded lattice interval in the group is pseudocomplemented. These groups form a discriminator variety which has a model completion, namely the divisible locally pseudocomplemented abelian \(\ell\)-groups with dense boolean part. We have similar results forpseudocomplemented MV-algebras, the intervals of locally pseudocomplemented abelian \(\ell\)-groups, actually Heyting (Gödel) algebras and thus the algebraic semantics for the union of Łukasiewicz and Gödel-Dummett logic. Some applications follow on definability in these varieties and the corresponding logics (an expansion of abelian logic in the former case). These results are related to recent work by Metcalfe and Reggio.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Xavier Caicedo (Universidad de los Andes, Colombia) - sesion: Operator Algebras zoom: https://us02web.zoom.us/j/83110888853 organizadores: - nombre: Fernando Abadie mail: fabadie@cmat.edu.uy - nombre: Alcides Buss mail: alcides.buss@ufsc.br - nombre: Damián Ferraro mail: damian@cmat.edu.uy charlas: - titulo: Separated graphs and dynamics. abstract: |A separated graph is a pair \((E,C)\), where \(E\) is a directed graph, \(C=\bigsqcup _{v\in E^ 0} C_v\), and \(C_v\) is a partition of \(r^{-1}(v)\) (into pairwise disjoint nonempty subsets) for every vertex \(v\). In recent years, separated graphs have been used to provide combinatorial models of several structures, often related to dynamical systems. This can be understood as a generalization of the common use of usual directed graphs in symbolic dynamics. I will survey some of these developments, including the failure of Tarski's dichotomy in the setting of topological actions, the construction of a family of ample groupoids with prescribed type semigroup, and the modeling of actions on the Cantor set.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Pere Ara (Universitat Autònoma de Barcelona, España) - titulo: A non-commutative topological dimension abstract: |Due to the Gelfand duality, C*-algebras are regarded as non-commutative topological spaces. This approach has sparked fundamental ideas in the structure and classification theory of C*-algebras. In this talk, I will explain a notion of non-commutative covering dimension and its impact in the classification programme of C*-algebras.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Jorge Castillejos (Instituto de Ciencias Matemáticas ICMAT, España) - titulo: Isotropy algebras and Fourier analysis for regular inclusions abstract: |Given a regular inclusion \(A\subseteq B\) of C*-algebras, with \(A\) abelian, we will define the notion of ``isotropy algebra'' for any given point \(x\) in the spectrum of \(A.\) We will then compare this notion with the usual notion of isotropy groups in the context of groupoid C*-algebras, showing that the isotropy algebra relative to the full groupoid C*-algebra is naturally isomorphic to the full group C*-algebra of the corresponding isotropy group. Based on the slightly generalized notion of ``isotropy module'', we will then introduce a method to analyze the structure of \(B\) paralleling the classical Fourier analysis. This talk is based on the paper [Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids, arXiv:1901.09683 [math.OA] [v3] Tue, 1 Jun 2021], joint with David Pitts.
start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Ruy Exel (Universidade Federal de Santa Catarina, Brasil) - titulo: Toeplitz algebras of semigroups and their boundary quotients abstract: |In general there are many distinct C*-algebras generated by isometric representations of a left cancellative monoid \(P\). Two distinguished examples are the Toeplitz C*-algebra \(\mathcal{T}_\lambda(P)\) generated by the left regular representation of \(P\) on \(\ell^2(P)\), and its boundary quotient \(\partial\mathcal{T}_\lambda(P)\). I will report on recent work with M. Laca, in which we define a universal Toeplitz C*-algebra \(\mathcal{T}_u(P)\) for a submonoid of a group that is canonically isomorphic to Li's semigroup C*-algebra when independence holds and is the universal analogue of \(\mathcal{T}_\lambda(P)\) also when independence fails. I plan to focus mainly on boundary quotients, explaining why the covariance algebra of the canonical product system over \(P\) with one-dimensional fibres may be regarded as the universal analogue of \(\partial\mathcal{T}_\lambda(P)\). I will give a concrete presentation of the full boundary quotient using a new notion of foundation sets, and give sufficient conditions on \(P\) for \(\partial\mathcal{T}_\lambda(P)\) to be purely infinite simple. If time permits, I will also address faithfulness of representations at the level of Toeplitz algebras.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Camila Senhem (Victoria University of Wellington, Nueva Zelanda) - titulo: Lifts of completely positive maps abstract: |Let \(A\) and \(B\) be \(C^*\)-algebras, \(A\) separable and \(I\) an ideal in \(B\). We show that for any completely positive contractive linear map \(\psi\colon A\to B/I\) there is a continuous family \(\Theta_t\colon A\to B\), for \(t\in [1,\infty)\), of lifts of \(\psi\) that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If \(A\) and \(B\) carry continuous actions of a second countable locally compact group \(G\) such that \(I\) is \(G\)-invariant and \(\psi\) is equivariant, then the family \(\Theta_t\) can be chosen to be asymptotically equivariant.
If a linear completely positive lift for \(\psi\) exists, then we can arrange that \(\Theta_t\) is linear and completely positive for all \(t\in [1,\infty)\); this yields an equivariant version of the Choi-Effros lifting theorem. In the equivariant setting, if \(A\), \(B\) and \(\psi\) are unital, the existence of asymptotically linear unital lifts are only guaranteed if \(G\) is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.
This talk is based on joint work with Marzieh Forough and Klaus Thomsen.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Eusebio Gardella (Universidad de Münster, Alemania) - titulo: Continuous orbit equivalence and full groups of ultragraph C*-algebras abstract: |In this talk, we describe ultragraphs, their associated edge shift spaces (which generalize SFT for infinite alphabets), and their associated C*-algebras and groupoids. We present results regarding continuous orbit equivalence of Deaconu-Renault systems and full groups associated with groupoids. To finish, we describe how to apply these results for ultragraph algebras.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Daniel Gonçalves (Universidade Federal de Santa Catarina, Brasil) - titulo: Revisiting quantum mechanics via quantales and groupoid C*-algebras abstract: |It is well known that operator algebras are the mathematical language of algebraic quantum mechanics and algebraic quantum field theory. In particular, von Neumann algebras cater for a natural formalization of Heisenberg's ``picture'' of quantum mechanics, which rests on interpreting physical observables as being time-dependent self-adjoint operators, to which von Neumann explicitly added, still in his twenties, that physical measurements should correspond to projection operators, thus producing the first mathematically thorough account of the foundations of quantum theory. However, from a conceptual point of view deep problems remained. These, in essence, have led to the seemingly unstoppable proliferation of interpretations and modifications of quantum mechanics, despite which, to this day, the ``measurement problem'' remains largely unsolved. In this talk I describe ongoing work whose aim is to address this by means of a definition of space of measurements whose open sets correspond to the finite pieces of classical information that can be extracted during a measurement. The interplay between C*-algebras, locally compact étale groupoids, and quantales plays an important role. The main purpose of the talk is to explain the main ideas surrounding measurement spaces along with related facts about groupoid C*-algebras and Fell bundles.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Pedro Resende (Instituto Superior Técnico, Portugal) - titulo: Permanence properties of verbal products and verbal wreath products of groups abstract: |Given a family of groups, the direct sum and the free product provide useful ways of constructing new groups out of it. Verbal products of groups, which were defined in the fifties, give many new operations in groups which share some common features with the direct sum and the free product. Arguably, these products have not been studied much in recent years and they might have been overlooked in geometric and measurable group theory.
In this talk we will review these products, give some examples and show that several group theoretical properties that are of much interest in operator algebras like amenability, Haagerup property, property (T), exactness, soficity, Connes' embedability, etc. are preserved under some of these products. Afterwards, we will explain how to define verbal wreath products between two groups, (these are semi-direct products that resemble the restricted wreath product), and we will show that the Haagerup property, soficity, Connes' embeddability and other metric approximation properties are preserved under taking verbal wreath products. Finally we will give some applications of them.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Román Sasyk (Universidad de Buenos Aires, Argentina) - sesion: Teoría de Números zoom: https://us02web.zoom.us/j/88466883296 organizadores: - nombre: María de los Ángeles Chara mail: charamaria@gmail.com - nombre: Guillermo Mantilla mail: guillermoa.mantillas@konradlorenz.edu.co - nombre: Amalia Pizarro mail: amalia.pizarro@uv.cl charlas: - titulo: Serre's modular conjecture. abstract: |In this talk we will recall the formulation of Serre's modular conjecture, and we will explain how some strong new modularity results can be used to give a simplified proof of it. This is a joint work with Luis Dieulefait.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Ariel Pacetti (Universidade de Aveiro, Portugal) - titulo: Monogenic and binary number fields of small degree abstract: |We show that a positive proportion of cubic and quartic number fields are not monogenic, and not just for local reasons. We also show that a positive proportion of quartic number fields are not binary, i.e. they don't arise as the invariant order attached to a binary quartic form. To prove these results, we study rational points (and the lack thereof) in a family of elliptic curves. Joint work with Levent Alpoge and Manjul Bhargava.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Ari Shnidman (Einstein Institute of Mathematics, Israel) - titulo: The integral trace form and shape as complete invariants for real Sn number fields abstract: |In this talk we will discuss two invariants attached to a number field: the integral quadratic form associated to its trace paring and its shape. We prove that, as long as we restrict the ramification, these invariants are in fact strong enough to completely determine the isomorphism class of a field within the family of totally real Sn-number fields; as a byproduct of the method we are able to describe the automorphism group of the integral trace form for such fields. The chief ingredient in the proofs is a linear algebra gadget for quadratic forms we called “Casimir pairings”. They generalize the Casimir element from the theory of Lie algebras as well as the usual inner product of 1-forms in Riemannian geometry. For trace forms of number fields (and more generally of étale algebras), I will explain how the main usefulness of Casimir pairings lies in their functorial properties which make them compatible with both Galois-étale theory and base changes. This is joint work with Guillermo Mantilla-Soler.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Carlos A. Rivera (University of Washington, Estados Unidos) - titulo: Upper bounds on counting number fields abstract: |I will present a general upper bound for the number of number fields of fixed degree and bounded discriminant. The method refines that of Ellenberg and Venkatesh, and improves upon their bound and that of Couveignes. This is joint work with Robert Lemke Oliver.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Frank Thorne (University of South Carolina, Estados Unidos) - titulo: Sums of certain arithmetic functions over \(\mathbb{F}_q[T]\) and symplectic distributions abstract: |In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function \(d_k(f)\) over short intervals and over arithmetic progressions for the function field \(\mathbb{F}_q[T]\) to certain integrals over the ensemble of unitary matrices when \(q \rightarrow \infty\). We study two problems: the average over all the monic polynomials of fixed degree that yield a quadratic residue when viewed modulo a fixed monic irreducible polynomial \(P\), and the average over all the monic polynomials of fixed degree satisfying certain condition that is analogous to having an argument (in the sense of complex numbers) lying at certain specific sector of the unit circle. Both problems lead to integrals over the ensemble of symplectic matrices when \(q \rightarrow \infty\). We also consider analogous questions involving convolutions of the von Mangoldt function. This is joint work with Vivian Kuperberg.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Matilde Lalín (Université de Montréal, Canadá) - titulo: Congruences satisfied by eta quotients abstract: |The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results given by Ahlgren and Ono for the partition function and by Treneer for more general Fourier coefficients state the existence of infinitely many families of congruences. We give an algorithm for computing explicit instances of such congruences for eta-quotients, and we illustrate our method with a few examples.
Joint work with Nathan Ryan, Zachary Scherr and Stephanie Treneer.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Nicolás Sirolli (Universidad de Buenos Aires, Argentina) - titulo: p-adic asymptotic distribution of CM points abstract: |A CM point in the moduli space of complex elliptic curves is a point representing an elliptic curve with complex multiplication. A classical result of William Duke (1988), complemented by Laurent Clozel and Emmanuel Ullmo (2004), states that CM points become uniformly distributed on the moduli space when we let the discriminant of the underlying ring of endomorphisms of these elliptic curves go to infinity. Since CM points are algebraic, it is possible to study p-adic analogues of this phenomenon. In this talk I will present a description of the p-adic asymptotic distribution of CM points in the moduli space of p-adic elliptic curves. This is joint work with Ricardo Menares (PUC, Chile) and Juan Rivera-Letelier (U. of Rochester, USA).
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Sebastián Herrero (Pontificia Universidad Católica de Valparaíso, Chile) - titulo: Malle's Conjecture for octic D4-fields abstract: |We consider the family of normal octic fields with Galois group D4, ordered by their discriminant. In forthcoming joint work with Arul Shankar, we verify the strong form of Malle's conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove this and related results.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Ila Varma (University of Toronto) - sesion: Dinámica de grupos zoom: https://us02web.zoom.us/j/82770882789 organizadores: - nombre: Juan Alonso mail: juan@cmat.edu.uy - nombre: Nancy Guelman mail: nguelman@fing.edu.uy - nombre: Sebastián Hurtado mail: shurtados@uchicago.edu - nombre: Cristóbal Rivas mail: cristobal.rivas@usach.cl charlas: - titulo: Orbit equivalence and dynamical properties of minimal group actions on the Cantor set abstract: |Two topological dynamical systems \((X, T, G)\) and \((Y,S,\Gamma)\) are said to be (topologically) orbit equivalent if there exists a homeomorphism \(h:X\to Y\) sending \(T\)-orbits onto \(S\)-orbits. A general question in the framework of orbit equivalence is how the dynamical properties of a given system are related to its orbit equivalence class. In this talk we will discuss about what are the natural restrictions that arise from being orbit equivalent, in the case of minimal shift actions on the Cantor set given by countable amenable groups. We will also discuss about the problem of realization of Choquet simplices as sets of invariant measures of topological dynamical systems and see how this is connected to the problem of classification up to orbit equivalence.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Paulina Cecchi (Universidad de Chile, Chile) - titulo: Random Walks and CAT(0) Cube Complexes abstract: |Let G be a group acting on a finite dimensional CAT(0) cube complex X. By studying equivariant maps from the Furstenberg-Poisson boundary to the Roller boundary, we deduce a variety of phenomena concerning the push-forward of the random walk from G to an orbit in X. Under mild and natural assumptions, we deduce positivity of the drift, sublinear tracking, and a central limit theorem. Along the way we prove that regular elements are plentiful and establish a homeomorphism between the boundary of the contact graph of X with a special subset of the Roller boundary called the regular points. This is joint work with Jean Lécureux and Frédéric Mathéus.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Talia Fernós (The University of North Carolina at Greensboro, Estados Unidos) - titulo: Optimal regularity of mapping class group actions on the circle abstract: |We prove that for each finite index subgroup \(H\) of the mapping class group of a closed hyperbolic surface, and for each real number \(r>0\) there does not exist a faithful \(C^{1+r}\)--action of \(H\) on a circle. (Joint with Thomas Koberda and Cristobal Rivas).
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Sang-hyun Kim (Korea Institute for Advanced Study, Corea del Sur) - titulo: Hyperbolic groups acting on their boundaries abstract: |A hyperbolic group acts on its Gromov boundary by homeomorphisms. In recent joint work with Jason Manning, we show these actions are topologically stable whenever the boundary is a sphere: any small perturbation of the action is semi-conjugate to the original action. This is also true for free groups, with cantor set boundary, and in ongoing work we are investigating the general case. In my talk, I will explain some of the strategy of the proof and motivation for the problem.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Kathryn Mann (Cornell University, Estados Unidos) - titulo: 'Locally moving groups acting on the real line I: C^1 actions' abstract: |Given a group G, we are interested in describing the possible actions of G on the real line, that is its representations into the group of homeomorphisms of the real line. In this series of two talks, we present a setup that allows to give a satisfactory picture for a class of groups, which arise as sufficiently rich subgroups of the group of homeomorphisms of the line, called locally moving groups. This class contains various finitely generated groups acting on the line: a famous example is Thompson’s group F.
In this first talk, we will focus on action by C^1-diffeomorphisms. In particular we will see that if G is a locally moving group of homeomorphisms on the line, then every faithful minimal action of G on the line is topologically conjugate to its natural defining action.
In the second talk, we will focus on actions that are only by homeomorphisms, that turn out to be much more flexible: for example we will explain that Thompson’s group F admits a continuum of exotic minimal faithful C^0 actions on the line. Nevertheless we will see that such actions have a very special structure: this will lead us to introduce the notion of R-focal action, a class of actions on the line that can be suitably encoded by certain actions on planar real trees. As an application we will obtain a local rigidity result: for a vast class of locally moving groups, all small perturbations of the natural defining actions are semi-conjugate to it.
The talks are based on joint work of the speakers with J. Brum and C. Rivas.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Michele Triestino (Université de Bourgogne, Francia) - titulo: 'Locally moving groups acting on the real line II: exotic actions' abstract: |Given a group G, we are interested in describing the possible actions of G on the real line, that is its representations into the group of homeomorphisms of the real line. In this series of two talks, we present a setup that allows to give a satisfactory picture for a class of groups, which arise as sufficiently rich subgroups of the group of homeomorphisms of the line, called locally moving groups. This class contains various finitely generated groups acting on the line: a famous example is Thompson’s group F.
In this first talk, we will focus on action by C^1-diffeomorphisms. In particular we will see that if G is a locally moving group of homeomorphisms on the line, then every faithful minimal action of G on the line is topologically conjugate to its natural defining action.
In the second talk, we will focus on actions that are only by homeomorphisms, that turn out to be much more flexible: for example we will explain that Thompson’s group F admits a continuum of exotic minimal faithful C^0 actions on the line. Nevertheless we will see that such actions have a very special structure: this will lead us to introduce the notion of R-focal action, a class of actions on the line that can be suitably encoded by certain actions on planar real trees. As an application we will obtain a local rigidity result: for a vast class of locally moving groups, all small perturbations of the natural defining actions are semi-conjugate to it.
The talks are based on joint work of the speakers with J. Brum and C. Rivas.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Nicolás Matte Bon (Université de Lyon, Francia) - titulo: Projective manifolds, hyperbolic manifolds and the Hessian of Hausdorff dimension abstract: |Let \(\Gamma\) be the fundamental group of a closed (real) hyperbolic \(n\)-manifold \(M.\) We study the second variation of the Hausdorff dimension of the limit set of convex co-compact morphisms acting on the complex-hyperbolic space \(\rho:\Gamma\to Isom(\mathbb H^n_\mathbb C)\), obtained by deforming a discrete and faithful representation of \(\Gamma\) that preserves a totally geodesic (and totally real) copy of the real-hyperbolic space \(\mathbb H^n_\mathbb R\subset\mathbb H^n_\mathbb C\). This computation is based on the study of the space of convex projective structures on \(M\) and a natural metric on it induced by the Pressure form. This is joint work with M. Bridgeman, B. Pozzetti and A. Wienhard.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Andrés Sambarino (Sorbonne Université, Francia) - sesion: Functional Differential Equations and its Applications zoom: https://us02web.zoom.us/j/82639200096 organizadores: - nombre: Pablo Amster mail: pamster@dm.uba.ar - nombre: Gonzalo Robledo mail: grobledo@uchile.cl charlas: - titulo: Coupled reaction-diffusion and difference system with nonlocal dispersal term and implicit time delay abstract: |We consider a class of biological models represented by a coupled system between reaction-diffusion and difference equations (renewal equation) with nonlocal dispersal term and implicit time delay. The difference equation generally arises, using the characteristic method, from an age-structured partial differential system. We start by studying the existence, uniqueness, positivity, and boundedness of solutions. We then investigate the stability analysis and obtain a threshold condition for the global asymptotic stability of the trivial steady state by using a Lyapunov functional. We also obtain a sufficient condition for the existence and uniqueness of a positive steady-state by using the method of lower and upper solutions. In the case of an unbounded domain, we study the existence of monotonic planar traveling wave fronts connecting the extinction state to the uniform positive state. The corresponding minimum wave speed is also obtained. In addition, we investigate the effect of the parameters on this minimum wave speed and we give a detailed analysis of its asymptotic behavior.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Mostafa Adimy (Institut de la Recherche en Informatique et Automatique, Francia) - titulo: Periodic positive solutions of superlinear delay equations via topological degree abstract: |We present a joint work with Pablo Amster (Universidad de Buenos Aires, Argentina) and Julián Haddad (Universidade Federal de Minas Gerais, Brazil), which deals with an extension to delay problems of some recent results by G. Feltrin and F. Zanolin. They obtained existence and multiplicity results for positive periodic solutions to nonlinear differential equations of the form \[u''(t)=f(t,u(t),u'(t))\] with periodic or Neumann boundary conditions and special assumptions on $f$. We obtain analogous results for a delay problem of the type \[u''(t)=f(t,u(t),u(t-\tau),u'(t))\] Our approach, as well as that of Feltrin and Zanolin, is topological and based on the coincidence degree introduced by J. Mawhin.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Pierluigi Benevieri (Universidade de Sao Paulo, Brasil) - titulo: A Krasnoselskii relatedness principle for delay differential equations abstract: |In this talk, we shall present an \textbf{Abstract} formulation of a duality principle established by Krasnoselskii. Under appropriate conditions, it shall be shown that, if the solutions of a nonlinear functional equation can be obtained by finding fixed points of certain operators in possibly different Banach spaces, then these operators have the same topological index. An application to delay differential equations shall be given. This is a joint work with P. Amster.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Julian Epstein (Universidad de Buenos Aires, Argentina) - titulo: Stability and boundedness of solutions of retarded equations abstract: |Generalized ODEs were introduced by J. Kurzweil in 1957 and are known to encompass several other types of equations. In this talk, we explore converse Lyapunov theorems for this type of equations. In particular, we derive results for certain integral forms of measure functional differential equations. We also relate Lyapunov stability to the boundedness of solutions of these equations which have non--absolute integrable right--hand sides.
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Márcia Federson (Universidade de Sao Paulo, Brasil) - titulo: Global bifurcation-like Theorems in presence of non-vanishing Spectral Flow abstract: |Given a one-parameter family of functions \(f_t:H \to \mathbb R\) where \(f_t (0) = 0, \nabla f_t (0) = 0\) and \(H\) is a real Hilbert space, the Spectral Flow of the Hessian of \(f\) was related recently to local bifurcation results by Fitzpatrick, Pejsachowicz and Waterstraat. While this invariant is finer than the topological index, the existent results are of local nature, in contrast to the global bifurcation theorem of Krasnoselski and Rabinowitz. We prove a global bifurcation theorem for “target values” of \(f\) under Spectral Flow hypothesis.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Julian Haddad (Universidade Federal de Minas Gerais, Brasil) - titulo: On the Stability and the Existence and Non-existence of \(T-\)Periodic Solutions for Nonlinear Delayed Differential Equations with Friction and \(\varphi\)-Laplacian abstract: |Let us consider the following problem \[(\varphi(x'(t) ))' + h(x(t),x'(t))x'(t)+ g(x(t-r)) = p(t), \ \ \ t\in [0,\infty),\] where \(\varphi:\mathbb{R} \rightarrow \mathbb{R}\) is an increasing homeomorphism such that \(\varphi(0)=0\), \(r\) is a positive constant, \(g:\mathbb{R} \rightarrow \mathbb{R}\) is a continuous differentiable function, and \(h:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}\) and \(p:[0,\infty) \rightarrow \mathbb{R}\) are continuous functions such that \(p\) is \(T\)-periodic in \(t\) for \(T\) a positive constant.
Using Lyapunov-Krasovskii functional and under appropriate assumptions we obtain new results on the global stability, boundedness of solutions, existence and non-existence of \(T-\)periodic solutions. This is a joint work with P. Amster and D.P. Santos.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Mariel Paula Kuna (Universidad de Buenos Aires, Argentina) - titulo: Linearized instability for neutral FDEs abstract: |In this talk, we will give a brief overview of a class of equations called neutral functional differential equations with state-dependent delays, describing some important applications. After this, we will show some recents results in the area, and we will present a principle of linearized instability for these equations.This is a joint work with Professor Bernhard Lani-Wayda.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Jaqueline Godoy Mesquita (Universidade de Brasilia, Brasil) - titulo: Existence of bifurcation point for impulsive differential equations via generalized ordinary differential equations abstract: |This is a joint work with Professors M. Federson and J. Mawhin. In this work, we establish conditions for the existence of a bifurcation point with respect to the trivial solution of a generalized ordinary differential equation, whose integral form displays the nonabsolute Kurzweil integral. The main tools employed here are the coincidence degree theory and an Arzela-Ascoli-type theorem for regulated functions. We also present applications to impulsive differential equations.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Maria Carolina Mesquita (Universidade de Sao Paulo, Brasil) - titulo: A Lazer-Leach type result for functional-differential equations at resonance abstract: |In this talk we will discuss some recent results regarding Lazer-Leach type conditions for the existence of periodic solutions to systems of functional-differential equations at resonance. We consider even-dimensional kernels and general delays using Coincidence Degree Theory.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Arturo Sanjuán (Universidad Distrital Francisco José de Caldas, Colombia) - titulo: On the nonlinearly determined wavefronts for the Mackey-Glass type diffusive equations abstract: |We study the Mackey-Glass type monostable delayed reaction-diffusion equation with a unimodal birth function \(g(u)\). This model, designed to describe evolution of single species populations, is considered in the presence of the weak Allee effect (\(g(u_0)>g'(0)u_0\) for some \(u_0>0\)). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. By considering the particular case of Nicholson's diffusive equation, we also discuss other situation leading to the appearance of nonlinearly determined wavefronts. This is a joint work with Karel Hasík, Jana Kopfová and Petra Nábělková, Mathematical Institute, Silesian University, Czech Republic.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Sergei Trofimchuk (Universidad de Talca, Chile) - sesion: Problemas Variacionales y Ecuaciones Diferenciales Parciales zoom: https://us02web.zoom.us/j/87857871842 organizadores: - nombre: Judith Campos Cordero mail: judith@ciencias.unam.mx - nombre: Duvan Henao Manrique mail: dhenao@mat.puc.cl - nombre: Dora Cecilia Salazar Lozano mail: dcsalazarl@unal.edu.co charlas: - titulo: Regularity for \(C^{1,\alpha}\) interface transmission problems abstract: |Transmission problems originally arose in elasticity theory, and they are nowadays of great interest due to its many applications in different areas in science. Typically, in these problems, there is a fixed interface where solutions may change abruptly, and the primary focus is to study their behavior across this surface. In this talk, we will discuss existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with \(C^{1,\alpha}\) interfaces. For this, we develop a novel geometric stability argument based on the mean value property.
These results are part of my PhD dissertation, and they are joint work with Luis A. Caffarelli and Pablo R. Stinga.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: María Soria-Carro (University of Texas, Estados Unidos) - titulo: A Non-variational Approach for the Porous Medium Equation abstract: |We extend the Krylov-Safonov theory and the corresponding H\"older estimates for the viscosity solutions of a type porous medium equation with non-variational structure. This work is in collaboration with Makson Santos (CIMAT-Guanajuato).
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Héctor Andrés Chang Lara (Centro de Investigación en Matemáticas, México) - titulo: Blow up for the Keller-Segel system in the critical mass case abstract: |We consider the Keller-Segel system in the plane with an initial condition with suitable decay and mass \(8 \pi\), which corresponds to the threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function \(u_0\) with mass \(8 \pi\) such that for any initial condition sufficiently close to \(u_0\) and the same mass, the solution is globally defined and blows-up in infinite time. We also find the profile and rate of blow-up. This result answers affirmatively the question of the nonradial stability raised by Ghoul and Masmoudi (2018). This is a joint work with M. del Pino, M. Musso and J. Wei.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Juan Dávila (University of Bath, Inglaterra y Universidad de Antioquia, Colombia) - titulo: Critical polyharmonic systems and optimal partitions abstract: |In this talk, we establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in \(\mathbb{R}^N\) which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to \(-\infty\). We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of \(\mathbb{R}^N\). Moreover, we give a detailed description of the shape of these domains.
This is a joint work with Mónica Clapp and Alberto Saldaña.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Juan Carlos Fernández Morelos (Universidad Nacional Autónoma de México, México) - titulo: Blowing up solutions for sinh-Poisson type equations on pierced domains abstract: |We consider sinh-Poisson type equations (either Liouville or mean field form) with variable intensities and Dirichlet boundary condition on a pierced domain \(\Omega_\varepsilon:=\Omega\setminus \cup_{i=1}^m B(\xi_i,\varepsilon_i)\), where \(\Omega\) is a smooth bounded domain in \(\text{I\!R}^2\) which contains the points \(\xi_i\), \(i=1,\dots,m\), \(\varepsilon_i=\varepsilon_i(\varepsilon)>0\), \(i=1,\dots,m\), depending on some parameter \(\varepsilon>0\) small enough. Given \(m\) different points \(\xi_i\), \(i=1,\dots,m\), we have found suitable radii \(\varepsilon_i\), \(i=1,\dots,m\) such that for \(\varepsilon>0\) small enough our problem (either Liouville or mean field form) has a solution \(u_\varepsilon\) blowing up positively around each \(\xi_1,\dots,\xi_{m_1}\) and negatively around \(\xi_{m_1+1},\dots,\xi_m\) respectively, as \(\varepsilon\to0\). We have used a family of solutions of the singular Liouville equation \[ \Delta u+|x -\xi|^{\alpha-2}e^{u}=0 \quad\text{ in $\text{I\!R}^2$, }\quad\text{satisfying}\qquad \displaystyle\int_{\text{I\!R}^2} |x -\xi |^{\alpha-2} e^u <+\infty \] to construct an approximation of the solution depending on some parameters, suitable projected and scaled in order to make the error small enough in a suitable norm. Hence, we have found an actual solution as a small additive perturbation of this initial approximation by using a fixed point argument.
Part of these results have been obtained in collaboration with Pierpaolo Esposito (Università di Roma Tre, Italia) and Angela Pistoia (Università di Roma ``La Sapienza'', Italia).
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Pablo Figueroa (Universidad Austral de Chile, Chile) - titulo: Non Linear Mean Value Properties for Monge-Ampère Equations abstract: |In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized \(p-\)Laplacian discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game.
Our goal in this talk is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Amp\`ere equation.
Joint work with P. Blanc (Jyväskylä), F. Charro (Detroit), and J.J. Manfredi (Pittsburgh).
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Julio D. Rossi (Universidad de Buenos Aires, Argentina) - titulo: Radially symmetric solutions to quasilinear problems with indefinite weight abstract: |In this talk we present recent results regarding the existence of infinitely many radially symmetric solutions to \[\label{eq main problem} \left\{ \begin{aligned} - \Delta_p u =&W(| x|)g(u) &\hbox{in} &\quad \,\,\,B_1 (0)\subset \mathbb{R} ^N, \,\\ u =& 0 & \hbox{in} & \quad \partial B_1 (0), \end{aligned} \right. \] where \(N\geq 2\), \(p>1\), \(\Delta _p u= \text{div} (|\nabla u|^{p-2} \nabla u)\) is the p-Laplacian operator, \(B_1 (0)\) stands for the unit ball in \(\mathbb{R} ^N\) centered at the origin, \(g: \mathbb{R} \longrightarrow \mathbb{R} \) is a nonlinear function which satisfies p-superlinearity conditions and the weight \(W: [0,1]\longrightarrow \mathbb{R}\) is a \(C^1 -\)function that changes sign.
Since the weight \(W\) takes negative values, the solutions to initial value problems associated to eq main problem may blow up at some \(r \in (X,1)\) and this fact represents a major difficulty when using phase-plane analysis arguments. We will discuss how this difficulty can be surpassed.
This is a recent work in collaboration with Alfonso Castro, Jorge Cossio and Sigifredo Herrón.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Carlos Vélez (Universidad Nacional de Colombia, Colombia) - titulo: 'A nonlocal isoperimetric problem: density perimeter' abstract: |We will discuss a variant of a classical geometric minimization problem in arbitrary dimensions \(d\geq 2\), known as nonlocal isoperimetric problem, which arises from studies in Nuclear Physics by Gamow in the 1930’s: \[ \inf_{\substack{\Omega\subset\mathbb{R}^d\\|\Omega|=m}}E_{a}^{\gamma}(\Omega):=\int_{\partial^*\Omega}a(x)d\mathcal{H}^{d-1}+\gamma\iint_{\Omega\times\Omega}\frac{1}{|x-y|^{\alpha}}dxdy \] for any \(\alpha\in(0,d)\) and \(\gamma>0\). By introducing a density \(a:\mathbb{R}^d\to[0,\infty)\) in the perimeter functional, we obtain features that differ substantially from existing results in the framework of the classical problem without densities. Firstly, we get existence of minimizing sets for a general class of coercive densities \(a(x)\) and all values of ``mass". In addition, we prove essential boundedness of such minimizers. Finally, in the regime of “small” non-local contribution we completely characterize the minimizer, for densities that are monomial radial weights. This work is a collaboration with Stan Alama and Lia Bronsard (McMaster University) and Ihsan Topaloglu (Virginia Commonwealth University), as part of the project QUALITATIVE PROPERTIES OF WEIGHTED AND ANISOTROPIC VARIATIONAL PROBLEMS financed by ANID CHILE FONDECYT INICIACION No. 11201259.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Andres Zuñiga (Universidad de O’Higgins, Chile) - sesion: Stochastic processes and applications zoom: https://us02web.zoom.us/j/89403082806 organizadores: - nombre: Joaquín Fontbona mail: fontbona@dim.uchile.cl - nombre: Juan Carlos Pardo mail: jcpardo@cimat.mx - nombre: Daniel Remenik mail: dremenik@dim.uchile.cl - nombre: Víctor Rivero mail: rivero@cimat.mx charlas: - titulo: Lattice trees in high dimensions abstract: |Lattice trees is a probabilistic model for random subtrees of \(\mathbb{Z}^d\). In this talk we are going to review some previous results about the convergence of lattice trees to the "Super-Brownian motion" in the high-dimensional setting. Then, we are going to show some new theorems which strengthen the topology of said convergence. Finally, if time permits, we will discuss the applications of these results to the study of random walks on lattice trees.
Joint work with A. Fribergh, M. Holmes and E. Perkins.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Manuel Cabezas (Pontificia Universidad Católica, Chile) - titulo: Large deviations for the exclusion process with slow boundary abstract: |We prove a large deviations principle for the empirical measure of the one-dimensional symmetric simple exclusion process with slow boundary. The name "slow boundary" comes from the fact that the dynamics at the boundary is slowed down with respect to the dynamics of the system. The hydrodynamic limit for this model was proved in [1], where the authors get that the boundary conditions of the hydrodynamic equations depend on the intensity of the rate at the boundary of the microscopic model. In the present work, [2], we study the non-critical case. This is joint work with Tertuliano Franco (UFBA) and Patrícia Gonçalves (IST).
References:
[1] Baldasso, R., Menezes, O., Neumann, A. et al. Exclusion Process with Slow Boundary. J Stat Phys 167, 1112–1142 (2017). https://doi.org/10.1007/s10955-017-1763-5
[2] Franco, T., Gonçalves, P., Neumann, A. (2021): Large Deviations for the SSEP with slow boundary: the non-critical case. Online: arxiv.
En esta charla daremos un resumen panorámico sobre Procesos de Lévy Libres incluyendo unimodalidad y comportamiento en tiempos cercanos a 0 y en infinito. Concluiremos hablando de resultados recientes encontrados con Hasebe para el caso multiplicativo en R+ y en el círculo unitario.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Octavio Arizmendi (Centro de Investigación en Matemáticas, México) - titulo: PageRank Nibble on directed stochastic block models abstract: |This talk will focus on the probabilistic analysis of the PageRank algorithm on directed stochastic block models with K communities. Specifically, we show that for sparse graphs, i.e., with stochastically bounded degrees, the distribution of a randomly chosen vertex within a given community converges in a Wasserstein metric as the number of vertices grows to infinity. Moreover, the set of limiting distributions for typical vertices in each of the K communities can be characterized via a system of branching stochastic fixed-point equations (SFPEs). We then show how this characterization via SFPEs can be used to analyze the PageRank Nibble algorithm, which is a version of personalized PageRank that can be used for community detection provided one has a seed set of vertices known to belong to the community of interest. Our approach provides a good simple heuristic for choosing the optimal damping factor and provides computable bounds for the probability of misclassification.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Mariana Olvera-Cravioto (University of North Carolina at Chapel Hill, EEUU) - titulo: Sharp bounds for consensus-based optimization of convex functions abstract: |Suppose one wants to optimize a function \(f:{\bf R}^d\to{\bf R}\). However, the function is not directly available, and all one can do is to use a ``black box" that on a selected input \(x\) produces the output \(f(x)\). Consensus-based optimization is a collection of techniques for optimizing \(f\) in this black box setting which is based on mean-field particle systems. We study a variant of this general method in the setting where \(f\) is strongly convex and smooth, and obtain sharp convergence bounds. This is work in progress with Dyego Araújo (IMPA).
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Roberto Imbuzeiro Oliveira (Instituto de Matemática Pura e Aplicada, Brasil) - titulo: Condensating zero-range process with condensation abstract: |Zero-range processes are interacting particle models where particles jump between different sites according to rates that only depend on the number of particles at the current position. In this talk we will focus on the case when the rates decrease with the number of particles, inducing a condensation phenomenon where a macroscopic number of particles occupies the same position. I will present recent results that identify the coarsening dynamics of the process as fluid limit. These techniques can be applied to derive fluid limits for Jackson networks with rates that depend on the number of customers in the queue. Joint work with Johel Beltrán, Daniela Cuesta, Milton Jara and Matthieu Jonckheere.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Inés Armendáriz (Universidad de Buenos Aires, Argentina) - titulo: Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance abstract: |This talk presents recent results on the cutoff phenomenon for a general class of asymptotically exponentially stable Ornstein-Uhlenbeck systems under ε-small additive Lévy noise. The driving noise processes include Brownian motion, α-stable Lévy flights, finite intensity compound Poisson processes and red noises and may be highly degenerate. Window cutoff thermalization is shown under generic mild assumptions, that is, we see an asymptotically sharp ∞/0-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure με along a time window centered in a precise ε-dependent time scale tε . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of the drift matrix Q. With this piece of theory at hand we provide a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to ε-small Brownian motion or α-stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.
Joint work with Gerardo Barrera (University of Helsinki, Finland) and Juan Carlos Pardo (CIMAT, México).
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Michael Hoegele (Universidad de los Andes, Colombia) - titulo: Scaling limit of the discrete Coulomb gas abstract: |The discrete Coulomb gas is a model where charged particles are put on the \(d\)-dimensional grid. In this talk, I will discuss the basic properties of the Coulomb gas, its connection with other statistical physics models and the scaling limit of the potential of the discrete Coulomb gas at high enough temperature. Joint work with Christophe Garban.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Avelio Sepúlveda (Universidad de Chile, Chile) - sesion: Large Stochastic systems zoom: https://us02web.zoom.us/j/86785073323 organizadores: - nombre: Claudio Landim mail: landim@impa.br - nombre: Pablo Ferrari mail: pferrari@dm.uba.ar charlas: - titulo: From stochastic dynamics to fractional PDEs with several boundary conditions abstract: |In this seminar I will describe the derivation of certain laws that rule the space-time evolution of the conserved quantities of stochastic processes. The random dynamics conserves a quantity (as the total mass) that has a non-trivial evolution in space and time. The goal is to describe the connection between the macroscopic (continuous) equations and the microscopic (discrete) system of random particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling; while the latter is a collection of particles that move randomly according to a transition probability and rings of Poisson processes. I will focus on a model for which we can obtain a collection of (fractional) reaction-diffusion equations given in terms of the regional fractional Laplacian with different types of boundary conditions.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Patrícia Gonçalves (Instituto Superior Técnico, Portugal) - titulo: Persistence phenomena for large biological neural networks abstract: |We study a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of biological mechanisms. We focus here on local interactions: upon firing, a given neuron increases the potential of a fixed number of randomly chosen neurons. We show a phase transition in terms of the stationary distribution of the limiting network. Whereas a finite network activity always vanishes in finite time, the infinite network might converge to either a trivial stationary measure or to a nontrivial one. This allows to model the biological phenomena of persistence: we prove that the network may retain neural activity for large times depending on certain global parameters describing the intensity of interconnection.
Joint work with: Maximiliano Altamirano, Roberto Cortez, Lasse Leskela.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Matthieu Jonckheere (Universidad de Buenos Aires, Argentina) - titulo: Structure recovery for partially observed discrete Markov random fields on graphs abstract: |Discrete Markov random fields on graphs, also known as graphical models in the statistical literature, have become popular in recent years due to their flexibility to capture conditional dependency relationships between variables. They have already been applied to many different problems in different fields such as Biology, Social Science, or Neuroscience. Graphical models are, in a sense, finite versions of general random fields or Gibbs distributions, classical models in stochastic processes. This talk will present the problem of estimating the interaction structure (conditional dependencies) between variables by a penalized pseudo-likelihood criterion. First, I will consider this criterion to estimate the interaction neighborhood of a single node, which will later be combined with the other estimated neighborhoods to obtain an estimator of the underlying graph. I will show some recent consistency results for the estimated neighborhood of a node and any finite sub-graph when the number of candidate nodes grows with the sample size. These results do not assume the usual positivity condition for the conditional probabilities of the model as it is usually assumed in the literature of Markov random fields. These results open new possibilities of extending these models to situations with sparsity, where many parameters of the model are null. I will also present some ongoing extensions of these results to processes satisfying mixing type conditions. This is joint work with Iara Frondana, Rodrigo Carvalho and Magno Severino.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Florencia Leonardi (Universidade de São Paulo, Brasil) - titulo: An algorithm to solve optimal stopping problems for one-dimensional diffusions abstract: |Considering a real-valued diffusion, a real-valued reward function and a positive discount rate, we provide an algorithm to solve the optimal stopping problem consisting in finding the optimal expected discounted reward and the optimal stopping time at which it is attained. Our approach is based on Dynkin's characterization of the value function. The combination of Riesz's representation of r-excessive functions and the inversion formula gives the density of the representing measure, being only necessary to determine its support. This last task is accomplished through an algorithm. The proposed method always arrives to the solution, thus no verification is needed, giving, in particular, the shape of the stopping region. Generalizations to diffusions with atoms in the speed measure and to non smooth payoffs are analyzed.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Ernesto Mordecki (Universidad de la República, Uruguay) - titulo: Branching processes with pairwise interactions abstract: |In this talk, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. BPI-processes form a subclass of branching processes with interactions, which were recently introduced by González Casanova et al. (2021), and includes the so-called logistic branching process which was studied by Lambert (2005).
Here, we provide a series of integral tests that fully explains how competition and cooperation regulates the long-term behaviour of BPI-processes. In particular, we give necessary and sufficient conditions for the events of explosion and extinction, as well as conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use the moment dual of BPI-processes which turns out to be a family of diffusions taking values on \([0,1]\), that we introduce as generalised Wright-Fisher diffusions together with a complete understanding of the nature of their boundaries.
start: 2021-09-16T17:30 end: 2021-09-17T18:15 speaker: Juan Carlos Pardo (Universidad Nacional Autónoma de México, México) - titulo: Exact solution of TASEP and generalizations abstract: |I will present a general result which allows to express the multipoint distribution of the particle locations in the totally asymmetric exclusion process (TASEP) and several related processes, for general initial conditions, in terms of the Fredholm determinant of certain kernels involving the hitting time of a random walk to a curve defined by the initial data. This scheme generalizes an earlier result for the particular case of continuous time TASEP, which has been used to prove convergence of TASEP to the KPZ fixed point. The result covers processes in continuous and discrete time, with push and block dynamics, as well as some extensions to processes with memory length larger than 1.
Based on joint work with Konstantin Matetski.
start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Daniel Remenik (Universidad de Chile, Chile) - titulo: Percolation on Randomly Stretched Lattices abstract: |In this talk we will give a new proof for a question that was first posed by Jonasson, Mossel and Peres, concerning percolation on a randomly stretched planar lattice. More specifically, we fix a parameter q in (0, 1) and we slash the lattice Z^2 in the following way. For every vertical line that crosses the x axis along an integer value, we toss an independent coin and with probability q we remove all edges along that line. Then we do the same with the horizontal lines that cross the y axis at integer values. We are then left with a graph G that looks like a randomly stretched version of Z^2 and on top of which we would like to perform i.i.d. Bernoulli percolation. The question at hand is whether this percolation features an non-trivial phase transition, or more precisely, whether p_c(G) < 1. Although this question has been previously solved in a seminal article by Hoffman, we present here an alternative solution that greatly simplifies the exposition. We also explain how the presented techniques can be used to prove the existence of a phase transition for other models with minimal changes to the proof.
This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Augusto Teixeira (Instituto de Matemática Pura e Aplicada, Brasil) - titulo: A martingale approach to lumpability abstract: |The martingale problem introduced by Stroock and Varadhan is an efficient method to prove the convergence of a sequence of stochastic processes which are derived from Markov processes. In this talk we present two examples to illustrate this approach: the density of particles per site of a sequence of condensing zero range processes and the number of sites occupied by a system of coalescing random walks evolving on a transitive finite graph. Both examples exhibit a sort of asymptotic lumpability.
start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Johel Beltrán (Pontificia Universidad Católica, Perú) - sesion: Geometría Algebraica computacional zoom: https://us02web.zoom.us/j/85818762305 organizadores: - nombre: Gregorio Malajovich mail: gregorio.malajovich@gmail.com - nombre: Diego Armentano mail: diego@cmat.edu.uy charlas: - titulo: On the number of roots of random invariant homogeneous polynomials abstract: |The number of roots of random polynomials have been intensively studied for a long time. In the case of systems of polynomial equations the first important results can be traced back to the nineties when Kostlan, Shub and Smale computed the expectation of the number of roots of some random polynomial systems with invariant distributions. Nowadays this is a very active field.
In this talk we are concerned with the variance and the asymptotic distribution of the number of roots of invariant polynomial systems. As particular examples we consider Kostlan-Shub-Smale, random spherical harmonics and Real Fubini Study systems.
start: 2021-09-13T16:45-0300 end: 2021-09-13T17:30-0300 speaker: Federico Dalmao Artigas (Universidad de la República, Uruguay) - titulo: Univariate Rational Sum of Squares abstract: |Landau in 1905 proved that every univariate polynomial with rational coefficients which is strictly positive on the reals is a sum of squares of rational polynomials. However, it is still not known whether univariate rational polynomials which are non-negative on all the reals, rather than strictly positive, are sums of squares of rational polynomials.
In this talk we consider the local counterpart of this problem, namely, we consider rational polynomials that are non-negative on the real roots of another non-zero rational polynomial. Parrilo in 2003 gave a simple construction that implies that if \(f\) in \(\mathbb R[x]\) is squarefree and \(g\) in \(\mathbb R[x]\) is non-negative on the real roots of \(f\) then \(g\) is a sum of squares of real polynomials modulo \(f\). Here, inspired by this construction, we prove that if \(g\) is a univariate rational polynomial which is non-negative on the real roots of a rational polynomial \(f\) (with some condition on \(f\) wrt \(g\) which includes squarefree polynomials) then it is a sum of squares of rational polynomials modulo \(f\).
Joint work with Bernard Mourrain (INRIA, Sophia Antipolis) and Agnes Szanto (North Carolina State University).
start: 2021-09-13T15:45-0300 end: 2021-09-13T16:30-0300 speaker: Teresa Krick (Universidad de Buenos Aires, Argentina) - titulo: On eigenvalues of symmetric matrices with PSD principal submatrices abstract: |Real symmetric matrices of size n, whose all principal submatrices of size \(k \lt n\) are positive semidefinite, form a closed convex cone. Such matrices do not need to be PSD and, in particular, they can have negative eigenvalues. The geometry of the set of eigenvalues of all such matrices is far from being completely understood. In this talk I will show that already when \((n,k)=(4,2)\) the set of eigenvalues is not convex.
start: 2021-09-14T15:00-0300 end: 2021-09-14T15:45-0300 speaker: Khazhgali Kozhasov (Technische Universität Braunschweig, Alemania) - titulo: Hausdorff approximation and volume of tubes of singular algebraic sets abstract: |I will discuss the problem of estimating the volume of the tube around an algebraic set (possibly singular) as a function of the dimension of the set and its degree. In particular I will show bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This problem is related to numerical algebraic geometry, where sets of ill-contidioned inputs are typically described by (possibly singular) algebraic sets, and their neighborhoods describe bad-conditioned inputs. Our result generalizes previous work of Lotz on smooth complete intersections in the euclidean space and of Bürgisser, Cucker and Lotz on hypersurfaces in the sphere, and gives a complete solution to Problem 17 in the book titled "Condition" by Bürgisser and Cucker. This is joint work with S. Basu.
start: 2021-09-13T15:00-0300 end: 2021-09-13T15:45-0300 speaker: Antonio Lerario (Scuola Internazionale Superiore di Studi Avanzati, Italia) - titulo: Algebraic computational problems and conditioning abstract: |We study the conditioning of algebraic computational problems. We have particular interest in which known theorems about conditioning on particular problems can be extended to the general context. We will pay special attention to the case in which the set of inputs with a common output is not a linear subspace.
start: 2021-09-14T15:45-0300 end: 2021-09-14T16:30-0300 speaker: Federico Carrasco (Universidad de la República, Uruguay) - titulo: Sparse homotopy, toric varieties, and the points at infinity abstract: |In my previous talk at CLAM-2021, I stated a complexity bound for solving sparse systems of polynomial equations in terms of condition numbers, mixed volume and surface and a few technical details. This results uses a technique that I call renormalization, and the bound excludes systems with solution at toric infinity.
In a recent paper, Duff, Telen, Walker and Yahl proposed a homotopy algorithm using Cox coordinates. This representation associates one new coordinate to each ray (one-dimensional cone) on the fan of the toric variety.
In this talk I will report on recent progress inspired by their work, on non-degenerate solutions at toric infinity for sparse polynomial systems.
start: 2021-09-14T16:45-0300 end: 2021-09-14T17:30-0300 speaker: Gregorio Malajovich (Universidade Federal do Rio de Janeiro, Brasil) - sesion: Análisis wavelet y aplicaciones zoom: https://us02web.zoom.us/j/86788957606 organizadores: - nombre: Victoria Vampa mail: victoriavampa@gmail.com - nombre: Liliam Alvarez Díaz mail: lilliam@ceniai.inf.cu - nombre: María Teresa Martín mail: mtmartin@fisica.unlp.edu.ar charlas: - titulo: Medición de la actividad eléctrica cardiaca utilizando algoritmos basados en la transformada wavelet abstract: |El electrocardiograma (ECG) es una señal que representa la actividad eléctrica del corazón. A partir de la digitalización del ECG se han implementado una diversidad de estudios que requieren algoritmos de medición automática. Algunos estudios son los registros Holter para monitoreo ambulatorio, el análisis del ECG de alta resolución, las pruebas de esfuerzo, monitoreo continuo en unidades coronarias o inclusive el monitoreo telemétrico en humanos y/o animales.
Como primera etapa se requiere la detección de las posiciones de inicios, máximos y fines de las ondas que constituyen la señal electrocardiográfica. Posteriormente, utilizando las posiciones previamente detectadas, se calculan duraciones, amplitudes e intervalos del ECG, utilizados para el monitoreo y/o diagnóstico cardiovascular.
El estado del arte presenta una gran diversidad de técnicas para detección, monitoreo y diagnóstico de registros electrocardiográficos digitales. Una de ellas se basa en el uso de la Transformada Wavelet (TW), siendo hasta el momento una de las herramientas más poderosas y con mayor rendimiento en el delineado del registro de ECG. Se presentaran las bases del algoritmo de delineado del ECG basado en la TW y resultados obtenidos de estudios en registros de humanos y animales.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Pedro D. Arini (Universidad de Buenos Aires, Argentina), joint with Javier G. Mendieta, and Santiago F. Caracciolo - titulo: Comportamiento de series de criptomonedas al inicio de la pandemia Covid-19 abstract: |Estudiamos el comportamiento de la memoria a largo plazo de rendimiento y volatilidad de siete de las criptomonedas más relevantes, durante un período que abarca antes y después del inicio de la pandemia. Se calcula el exponente de Hurst por un método wavelet que hemos modificado inspiradas en la bibliografía. Comparamos los resultados obtenidos en diferentes frecuencias, llegando a altas frecuencias, de muestreo y discutimos el efecto de la pandemia de Covid-19 sobre el rendimiento y la volatilidad.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Ma. Belén Arouxet (Universidad Nacional de La Plata, Argentina), joint with Verónica E. Pastor - titulo: Análisis de coherencia wavelet en finanzas con datos alternativos abstract: |Los mercados financieros son muy ricos en datos y permiten el estudio y modelización de procesos estocásticos de una gran importancia económica. Tradicionalmente, los estudios se realizaban con datos generados dentro de los mercados (precios, volúmenes negociados, etc.). Sin embargo, en los últimos años han surgido datos "alternativos". Entendemos por "alternativos" a aquellos datos que, si bien no surgen del propio mercado, pueden explicar el comportamiento de series temporales financieras: Twitts, consultas de Wikipedia, geolocalización de consumidores, etc.
Nosotros estudiamos la interacción de los precios y volatilidades de criptomonedas con la atención del público en dicho mercado medido por medio de Google Trends. Comparamos resultados utilizando diferentes conjuntos de palabras clave contenidas en Google Trends, y en distintos momentos del mercado (mercado bajista, alcista y plano).
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Aurelio F. Bariviera, (Universitat Rovira i Virgili, España) - titulo: Bases B-spline wavelet ortogonales en el intervalo en la solución de problemas de valores de contorno abstract: |Wavelets y análisis multirresolución constituyen una herramienta atractiva para la resolución numérica de ecuaciones diferenciales. La utilización de bases wavelet en los métodos wavelet-Galerkin permiten obtener esquemas numéricos eficientes y de elevada precisión.
En el marco de un Análisis Multirresolución (AMR) en el intervalo, se propone la construcción de una base B-spline wavelet con un requerimiento de ortogonalidad sobre sus derivadas entre distintas escalas de aproximación. La base está formada por wavelets interiores (que se obtienen de traslaciones y dilaciones una wavelet madre) cuyo soporte está contenido en el intervalo y wavelets de borde especialmente diseñadas. Al aplicar estas bases en la discretización de ecuaciones diferenciales de segundo orden, mediante esquemas del tipo wavelet-Galerkin, conducen a la resolución de sistemas lineales. Debido al soporte compacto y la condición de ortogonalidad requerida, las matrices asociadas tienen buenas propiedades, son ralas o esparcidas (diagonales por bloques y cada bloque es una matriz banda) con número de condición uniformemente acotado, lo que permite eficiencia en los cálculos y buenos resultados de convergencia a la solución con bajo costo computacional.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Lucila D. Calderón, (Universidad Nacional de La Plata, Argentina), joint with María T. Martín - titulo: 'Algoritmo para recuperar señales espacio-temporales con descomposición rala utilizando diccionarios wavelet: aplicación al mapeo electrocardiográfico' abstract: |La construcción de imágenes a partir del electrocardiograma de superficie es una técnica de diagnóstico médico utilizada para estimar, no invasivamente, el potencial eléctrico en la superficie del corazón (epicardio) a partir del potencial eléctrico en el torso del paciente.
Para llevar a cabo dicha técnica, se requiere 1) una imagen médica con la cual se puede crear un modelo del torso y del músculo cardíaco, 2) la adquisición no invasiva de una gran cantidad de canales electrocardiográficos. A partir de lo mencionado, puede estimarse el potencial eléctrico sobre el epicardio, a través de la solución a un problema inverso mal condicionado el cual debe ser regularizado.
En el presente trabajo se desarrolla un algoritmo de optimización rala (del inglés, sparse) diseñado para modelos lineales con estructura de Kronecker el cual fue penalizado con una mezcla entre la norma 21 (LASSO de grupo) y la norma 2 al cuadrado (Tikhonov). Nuestra hipótesis se basó en que el potencial epicárdico puede descomponerse de manera rala en diccionarios creados a partir de funciones y/o escalas wavelet.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Santiago F. Caracciolo (Universidad de Buenos Aires, Argentina) joint with César F. Caiafa, Francisco D. Martínez Pería and Pedro D. Arini - titulo: Detection and analysis of micro saccadic movements during reading using the Continuous Wavelet Transform abstract: |Reading requires the integration of several central cognitive subsystems from attention and oculomotor control to word identification and language comprehension. When reading, the eyes alternate between long movements and relative stillness, that are called saccadic movements and fixations, respectively. The average fixation lasts for 150 to 250 ms and it is composed by three movements called microsaccades (or microsaccadic movements), tremor and drift. Drift and tremor are slow movements with small amplitude; microsaccades represent a ballistic component of fixational eye movements. Then, microsaccades are characterized as roughly linear movement epochs with durations up to 30ms and a frequency of one to two per second in fixations not related with reading. They are considered as binocular movements with the standard definition of binocularity used in the literature. There are just a few works analyzing microsaccades while subjects are processing complex information and fewer when doing predictions about upcoming events. In all of them there is evidence that microsaccades are sensitive to changes of perceptual inputs as well as modulations of cognitive states. Changes in perceptual inputs are related to the type of sentences (low/high predictability, proverbs) and the characteristics of the words in the sentence (frequency, predictability, length, etc.). For this reason we think it is important to detect and characterise microsaccadics during the reading process.
It is well known that the Continuous Wavelet Transform (CWT) is an efficient method for displaying and analyzing characteristics of nonstationary signals that are dependent on time and scale and then on frequency. Taking this into account, it is possible to say that it provides a very useful tool for detecting and identifying particular spectral features of the analyzed signal, transient information content and the nonstationary properties, among others. In the context of eye movements, it has been used to characterize and extract microsaccades. Microsaccades can be modeled as smoothed singularities within a time series. These local singularities can be identified using, for example, the CWT and the results are not sensitive to the choice of the mother wavelet.
In time-frequency representation, a singularity at time \(t_0\) appears as a cone-like structure in the modulus of the wavelet transform. At lower frequencies, the wavelet is broader and the width of the cone scales with the width of the wavelet function at each voice or scale a. The wavelet focuses the whole variance at time points of singularities into the cone and its maximum at each voice corresponds to the position of the singularity in the time series. Therefore, the singularities can be detected using the modulus of the wavelet transform.
The method of maximum modulus lines is a numerical method for detecting singularities in time series using the CWT. A maximum modulus line is a line on the time-scale plane on which the modulus of the wavelet transform has a local maximum with respect to small variation of the translation parameter \(b_0\), that is $$ |W_{f,\psi}(a,b_0)|>|W_{f,\psi}(a,b_0\pm\epsilon)|.$$
As usual, \(W_{f,\psi}\) stands for the CWT of the time series of the signal \(f\) using the mother wavelet \(\psi\) and \(a, b_0\) the dilation and translation parameters, respectively. This points of maximum modulus are then connected giving as resulta the maximum modulus lines. It can be shown that if a signal has a singularity at a given point, then there is a maximum modulus line that converges towards the location of the singularity at small scales. In smoothed singularities the maximum modulus line may end at a scale which is approximately the smoothing scale of the singularity. For this reason, the maximum modulus lines, which go from a fixed highest frequency/smallest scale to a smallest frequency/largest scale were considered and the estimated position of the singularity is simply the small-scale end of the corresponding maximum modulus line.
In our work we have considered as mother wavelet the following function, $$\psi(t)=-\theta'(t), \qquad \theta(t)=\text{e}^{-t^2}.$$ This mother wavelet has a null moment and the wavelet transform of the signal has a similar behaviour to the derivative of the signal. In this way, the CWT allows to detect the abrupt changes of the position, i.e. the singularities of its velocity.
Another method for detecting microsaccades is to consider them as outliers of the velocity since, as we have said before, they can be thought as ballistic movements. For the algorithm used in this method of detection, it is necessary to determine the value of a parameter \(\lambda\) and a minimum duration for the movement to be considered as a microsaccadic. Different works use different values for these parameters since it depends on the application and the determined value so that the detection is coherent. For our work we determined that \(\lambda=1.5\) and a minimum duration of 6 samples (i.e. 5ms duration) turn out to be a mean rate according to different works related to reading experiments.
Comparing the method of maximum modulus lines and the velocity algorithm using these parameters we have precisely determined the microsaccadic movements during the reading process.
For the experiment, we have considered two different groups of healthy people with similar education: one group of young adults and other group of elderly people. The first group consisted of 40 adults with mean age 28 (SD=4.2 years) and mean education 18.2 (in years). The second one involved 40 adults with mean age 71 (SD=6.1 years), mean education 15.1 (in years). We only took into account data of 19 subjects from the first group and 18 from the second one. To perform the experiment, we used a Spanish sentence corpus composed of 76 regular sentences that represent a large variety of grammatical structures, also called Low Predictability Sentences, and 64 Proverbs that are common our Argentinian culture. Each sentence was displayed in the centerline of a 20-inch LCD Monitor (1024\(\times\)768 pixels resolution; font: regular New Courier, 18 point, vertical size of one character: 0.2 in height). The participants were seated in front of the monitor at a distance of 60 cm. Head movements were minimized using a chin rest. The participants eye movements were recorded with an EyeLink 1000 Desktop Mount (SR Research) eyetracker, with a sampling rate of 1000 Hz and an eye position resolution of 20-s arc. All recordings and calibration were binocular. The participants gaze was calibrated with a standard 13-point grid for both eyes. After validating the calibration, a fixation point appeared at the position where the first letter of the sentence was to be presented. As soon as both eyes were detected within a one grade radius relative to the fixation point, the sentence was presented. After reading it, participants had to move their eyes to a dot in the lower right corner of the screen to end the trial.
The sentences ranged from 5 to 14 words in length; mean length being 7.3 (SD=1.9) words. The words ranged from 1 to 14 letters in length; the mean word length was 4.0 (SD=2) letters. All the participants read the whole set of sentences but the data with registration or reading errors were discarded. So the group of young participants read 3385 sentences and the group of adults read 2569 sentences. We have first detected the fixations on the words and then extracted the microsaccadic movements. For the first group were counted 27594 fixations with mean duration of 206.83ms (SD=105.58) while for the second one were counted 22571 fixations with a mean duration of 203.80ms (SD=100.30).
Now let us recall the definition of maxjumpword (MJW): it is the word with the largest difference between the close predictability of two consecutive words and can be determined according to the following equation< \[ maxjumpword=\text{max}[\text{logit}(pred_{N+1}-\text{logit}(pred_{N}], \] being \(\text{logit}\big(pred_{N}\big)=0.5\hspace{2mm}\text{ln} \Big(\frac{pred_{N}}{1-pred_{N}}\Big)\) and \(pred_{N}\) the predictability of the read word \(N\).
In this work we extract the microsaccadics and perform a first analysis of changes in some of its characteristics respect to the relative position of the MJW. The considered characteristics are: length of the movement, vertical and horizontal variations, maximum velocity and duration.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Liliana R. Castro (Universidad Nacional del Sur, Argentina), joint with Juan M. Arriola and Marcela P. Álvarez - titulo: Entropía y Complejidad Wavelet para identificar la inestabilidad eléctrica cardíaca en pacientes con infarto de miocardio abstract: |Se ha demostrado que aquellos individuos que han sufrido un infarto de miocardio (IM) tienen una alta probabilidad de desarrollar arritmias ventriculares malignas y/o muerte súbita cardíaca. Las anomalías de la conducción eléctrica que aparecen en la región infartada del miocardio se reflejan en el electrocardiograma (ECG) como fragmentaciones del complejo QRS (fQRS). Dichas fragmentaciones no siempre son posibles de detectar visualmente. Si bien puede suceder que los pacientes no desarrollen taquicardia y/o fibrilación venticular (VT/VF), son potencialmente riesgosos porque pueden desarrollarla inesperadamente y sin síntomas previos.
Hay pocas técnicas no invasivas para capturar dichas inestabilidades eléctricas, como la técnica de electrocardiograma de señal promediada, y la varianza espectral del complejo QRS, entre otras.
Por ello, hemos evaluado la Entropía y Complejidad Wavelet del complejo QRS del ECG, utilizando la Transformada Wavelet Continua, como un método eficaz para cuantificar estas alteraciones anormales en la actividad eléctrica cardíaca en pacientes post IM que no presentan VT/VF.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Gisela Clemente, (Universidad Nacional de La Plata, Argentina), joint with Esteban Valverde - titulo: 'Wavelet Analysis in Space Research at the Applied Mathematics and Geophysics group at INPE: advances and overview' abstract: |This presentation aims at a comprehensive view of ongoing research efforts in multiscale analysis applied to challenges in the context of space environment and its electrodynamical effects on the Earth and nearby regions. Based on the developments by the INPE group LANCE and collaborators, the content explored takes an overview and the next steps on the multiscale research. We discussed some results obtained in the wavelet analysis of space and geophysical dataset and their implication on understanding the physics processes. Moreover, we present the first successful results obtained in the wavelet-based multiresolution regularity detection methodology with the generic block-structured mesh adaptation in evolutionary partial differential equations. In particular, the solvers that will permit fast and reliable space weather magneto-hydrodynamic simulations and their needs in the multiscale real time-space data modelling in future.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Margarete Oliveira Domingues (Instituto Nacional de Pesquisas Espaciais, Brasil), joint with Odim Mendes and Muller Moreira Lopes - titulo: 'Cryopreservation Process: analyzed through Wavelet Transforms' abstract: |Rapid and ultra rapid cooling methods, in which liquid nitrogen (\(LN_2\)) is used as cooling agent, require special equipment and methods in order to record and analyze the thermal history of the sample to be cryopreserved. The temperature profile during the process of cooling is an important parameter in the design and description of any protocol for cryopreservation. We present a home made device, used to record the temperature at high rate during the cooling in \(LN_2\). System noise at room temperature is about \(\pm ~ 2.9 ^{\circ}\)C and at \(LN_2\) is about \(\pm ~ 6.9 ^{\circ}\)C, when in the liquid nitrogen it is (\(-196^{\circ}\)C). This noise introduced when measuring the temperature may, in some cases, difficult the analysis and interpretation of the signal. A related use of Wavelet transform is for smoothing and or de noising data based on wavelet coefficients, by means of removing the undesired frequency components. With the intention of eliminate high frequencies noise, we apply a filter in such a way that in the level ``\(j\)'' of the process, the residuals will be a soften version of the original signal having less amount of high frequency signals in contrast with level ``\(j+1\)'', and half number of data. These are the main points by which, in the present work, we have used a signal isolation method based on orthogonal wavelets, in order to analyze the remaining signal with minimum modification of the associated dynamics. Using this filtering method, system noise was virtually eliminated allowing a more precise interpretation of the significance of cooling curves.
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Ana Korol (Universidad Nacional de Rosario, Argentina) - titulo: Aplicación de Transformada Wavelet Continua a señales de Emisión Acústica de fractura de materiales frágiles abstract: |La Emisión Acústica es un ensayo no destructivo que nos permite monitorear la integridad de un material frágil sometido a esfuerzos. Si, por ejemplo, un cilindro de roca es comprimido hasta su fractura se generan ondas elásticas en el material que a través de un sensor se convierten en ondas eléctricas, denominadas señales de Emisión Acústica. Estas señales nos dan información sobre los procesos complejos que se producen en el material tales como nucleación, crecimiento y coalescencia de microfacturas hasta que finalmente se producen macrofracturas que llevan a la ruptura del material. Las señales explosivas de EA que se generan durante el proceso involucran un amplio rango de tiempos y escalas por lo que a través del análisis con TWC nos permite identificar y evaluar los diferentes procesos.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Rosa Piotrkowski (Universidad de Buenos Aires, Argentina), joint with Miguel Eduardo Zitto - titulo: Uso de la Transformada Wavelet en el análisis de datos de la evolución del Covid-19 abstract: |Desde el comienzo de la pandemia originada por el SARS-COV 2, se han utilizado distintas herramientas para analizar los datos disponibles, a fin de comprender su dinámica. En gran medida, en el análisis, se hizo uso de modelos matemáticos diseñados para describir la evolución de la pandemia. Se aplicaron, además, técnicas de análisis de series temporales, las cuales tienen la ventaja de acumular información, al contar con un mayor volumen de datos con el paso del tiempo. El número de infectados y muertos diarios, entre otros datos, corresponden a series no estacionarias, dado que sus propiedades estadísticas varían con el tiempo. A modo de ejemplo, la cantidad de personas a infectarse tiende a reducirse a medida que más personas se infectan. Además, si el número de infectados aumenta, suelen tomarse diferentes medidas, tales como las que restringen la movilidad, al mismo tiempo que el avance de la vacunación tiende a producir una disminución de los casos. Todos estos factores, tomados de conjunto, le otorgan una dinámica compleja a las series a estudiar: en ellas conviven múltiples fenómenos localizados tanto en el tiempo como en frecuencia, que se superponen bajo complejas estructuras. La Transformada Wavelet se presenta como una herramienta apropiada para analizar series de este tipo, dado que puede ser aplicada a series no estacionarias, y permite estudiar las relaciones subyacentes entre sus datos en el tiempo y en frecuencia, así como encontrar patrones de comportamiento. En este trabajo aplicamos estas herramientas a las series de datos de infectados, fallecidos y de movilidad de la provincia de Buenos Aires y mostramos, apelando a distintos cuantificadores, tales como la coherencia wavelet, el análisis tiempo-frecuencia de las relaciones entre distintas series. De esta forma la Transformada Wavelet se coloca como una herramienta de análisis relevante a tener en cuenta en el proceso de toma de decisiones.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Victoria Vampa (Universidad Nacional de La Plata, Argentina), joint with Federico Holik - sesion: Geometría Algebraica zoom: https://us02web.zoom.us/j/85805078167 organizadores: - nombre: Álvaro Rittatore mail: alvaro@cmat.edu.uy - nombre: Pedro Luis del Ángel R. mail: luis@cimat.mx charlas: - titulo: Birational geometry of Calabi-Yau pairs abstract: |Recently, Oguiso addressed the following question, attributed to Gizatullin: ``Which automorphisms of a smooth quartic K3 surface \(D\subset \mathbb{P}^3\) are induced by Cremona transformations of the ambient space \(\mathbb{P}^3\)?'' When \(D\subset \mathbb{P}^3\) is a quartic surface, \((\mathbb{P}^3,D)\) is an example of a \emph{Calabi-Yau pair}, that is, a pair \((X,D)\) consisting of a normal projective variety \(X\) and an effective Weil divisor \(D\) on \(X\) such that \(K_X+D\sim 0\). In this talk, I will explain a general framework to study the birational geometry of mildly singular Calabi-Yau pairs. This is a joint work with Alessio Corti and Alex Massarenti.
start: 2021-09-14T15:00-0300 end: 2021-09-14T15:45-0300 speaker: Carolina Araujo (Instituto de Matemática Pura e Aplicada, Brasil) - titulo: On reconstructing subvarieties from their periods abstract: |We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. This is a joint work with Emre Sertoz.
start: 2021-09-15T16:45-0300 end: 2021-09-15T17:30-0300 speaker: Hossein Movasati (Instituto de Matemática Pura e Aplicada, Brasil) - titulo: 'Quantum cohomology and derived categories: the case of isotropic Grassmannians' abstract: |The relation between quantum cohomology and derived categories has been known since long ago. In 1998 Dubrovin formulated his celebrated conjecture relating semisimplicity of the quantum cohomology of Fano manifolds and the existence of exceptional collections in their derived categories of coherent sheaves. In this talk we will discuss recent joint work of the author with Anton Mellit, Nicolas Perrin and Maxim Smirnov studying the relation between quantum cohomology for certain isotropic Grassmannians and their quantum cohomology in the spirit of Dubrovin conjecture. In this case the small quantum cohomology turns out not to be semisimple but still an interesting decomposition of the derived category arises. If time permits new directions and some problems will be mentioned.
start: 2021-09-13T15:45-0300 end: 2021-09-13T16:30-0300 speaker: John Alexander Cruz Morales (Universidad Nacional de Colombia, Colombia) - titulo: Initial degeneration in differential algebraic geometry abstract: |By a degeneration, we mean a process that transforms a geometric object \(X/F\) defined over a field \(F\) into a simpler object that retains many of the relevant properties of \(X\). Formally, any degeneration is realized by an integral model for \(X\); that is, a flat scheme \(X'/R\) defined over some integral domain \(R\) whose generic fiber is the original object \(X\).
In this talk, we endow the field \(F=K((t_1,\ldots,t_m))\) of quotients of multivariate formal power series with a (generalized) non-Archimedean absolute value \(|\cdot|\). We use this to establish the existence of (affine) integral models \(X'/R\) over the ring of integers \(R=\{|x|\leq 1\}\) for schemes \(X\) associated to solutions of systems of algebraic partial differential equations with coefficients on \(F\). We also concretely describe the specialization map of a model \(X'/R\) to the maximal ideals of \(R\), which are encoded in terms of usual (total) monomial orderings.
start: 2021-09-13T16:45-0300 end: 2021-09-13T17:30-0300 speaker: Cristhian Garay López (Centro de Investigación en Matemáticas, México) - titulo: Rank two bundles over fibered surfaces abstract: |Let S be a smooth complex surface fibered over a curve C. Under suitable assumptions, if E is a stable rank two bundle on C then its pullback is a H-stable bundle on S. In this sense we can relate moduli spaces of rank two stable bundles on the surface S with moduli spaces of rank two stable bundles on C. In this talk we aim to take advantage of this relation to provide examples of Brill–Noether loci on fibered surfaces.
start: 2021-09-14T15:45-0300 end: 2021-09-14T16:30-0300 speaker: Graciela Reyes (Universidad Autónoma de Zacatecas, México) - titulo: Langlands Program and Ramanujan Conjecture abstract: |We will give an overview of several aspects of the Langlands Program that interconnect different fields of mathematics, namely: Algebraic Geometry, Number Theory and Representation Theory. We study general results over global fields, and mention what is known on Langlands functoriality conjectures. In characteristic p, we have a rich interplay with algebraic geometry and we present results of the author in connection with the work of Laurent and Vincent Lafforgue. As a main application, we look at the Ramanujan conjecture, known for generic representations of the classical groups and GL(n) over function fields, for example.
start: 2021-09-14T16:45-0300 end: 2021-09-14T17:30-0300 speaker: Luis Alberto Lomelí (Pontificia Universidad Católica, Chile) - titulo: Actions and Symmetries abstract: |I will discuss some classical and some recent results about group and algebra actions on abelian varieties and curves.
start: 2021-09-13T15:00-0300 end: 2021-09-13T15:45-0300 speaker: Rubí Rodríguez (Universidad de la Frontera, Chile) - titulo: Representation of groups schemes that are affine over an abelian variety abstract: |The classical representation theory of affine schemes over a field, has been widely considered and many of its main properties have been understood and put under contral--after the work of many mathematicians along the second half of the 20th century. In joint work with del Ángel and Rittatore, we proposed a representation theory for the more general situation where the group scheme is affine over a fixed abelian variety A (the classical case corresponds to the situation that A is the trivial abelian variety with only one point). In this talk we describe the main relevant definitions, where the representations are vector bundles over A with suitable actions of the group scheme G. Then, we show that in our context an adequate formulation of Tannaka--type of recognition and reconstruction results (due to Grothendieck, Saavedra, Deligne and others) remain valid.
start: 2021-09-15T15:45-0300 end: 2021-09-15T16:30-0300 speaker: Walter Ferrer (Universidad de la República, Uruguay) - titulo: A criterion for an abelian variety to be non-simple and applications abstract: |Let \((A,\mathcal{L})\) be a complex abelian variety of dimension \(g\) with polarization of type \(D = \mbox{diag}(d_1,\dots,d_g)\). So \(A = V/\Lambda\) where \(V\) is a complex vector space of dimension \(g\) and \(\Lambda\) is a lattice of maximal rank in \(\mathbb{C}^g\) such that with respect to a basis of \(V\) and a symplectic basis of \(\Lambda\), \(A\) is given by a period matrix \((D\; Z)\) with \(Z\) in the Siegel upper half space of rank \(g\).
In this talk we will present part of the results in Auffarth, Lange, Rojas (2017), where we give a set of equations in the entries of the matrix \(Z\) which characterize the fact that \((A,\\mathcal{L})\) is non-simple.
We developed this criterion to apply it in the problem of finding completely decomposable Jacobian varieties. This research is motivated by Ekedahl and Serre's questions in Ekedahl, Serre (1993), which can be summarized into whether there are curves of arbitrary genus with completely decomposable Jacobian variety. Moreover, in Moonen, Oort (2004), the authors ask about the existence of positive dimensional special subvarieties \(Z\) of the Jacobian loci \(\mathcal{T}_g\) in the moduli space of principally polarized abelian varieties, such that the abelian variety corresponding with the geometric generic point of \(Z\) is isogenous to a product of elliptic curves.
In this direction, we use this criterion in combination with the so called Group Algebra Decomposition Lange, Recillas (2004) of a Jacobian variety \(JX\) with the action of a group \(G\). This is, the decomposition of \(JX\) induced by the decomposition of \(\mathbb{Q}[G]\) as a product of minimal (left) ideals which gives an isogeny \[B_1^{n_1} \times \cdots \times B_r^{n_r}\to JX\]
Although \(JX\) is principally polarized, the induced polarization on \(B_i\) is in general not principal. The question we address is to study the simplicity of \(B_i\), hence finding in some cases a further decomposition of \(JX\).
This is a joint work with R. Auffarth and H. Lange.
start: 2021-09-15T15:00-0300 end: 2021-09-15T15:45-0300 speaker: Anita Rojas (Universidad de Chile, Chile) - sesion: K-teoría zoom: https://us02web.zoom.us/j/89144041460 organizadores: - nombre: Noé Bárcenas mail: barcenas@matmor.unam.mx - nombre: Eugenia Ellis mail: eellis@fing.edu.uy charlas: - titulo: K-teoría y problemas de clasificación abstract: |El formato general de los problemas de clasificación que nos ocupan es el siguiente. Se tienen un anillo de base \(\ell\), una familia \(C\) de \(\ell\)-álgebras, y un invariante \(K\)-teórico (e.g. \(K_0\) con su preorden natural), y nos preguntamos si dos integrantes de la familia con invariantes isomorfos son necesariamente isomorfas o equivalentes bajo algún criterio (e.g. Morita equivalentes). Una fuente de estos problemas es tomar una clase de \(C^*\)-álgebras (e.g. completaciones de \(\mathbb{C}\)-álgebras definidas por cierto tipo de generadores y relaciones) que ha sido clasificada por su \(K\)-teoria (topológica) y preguntarse si la \(K\)-teoría algebraica clasifica el análogo puramente algebraico de esa familia (e.g. la dada por el mismo tipo de generadores y relaciones, sin completar). En la charla veremos algunos ejemplos de problemas de ese tipo y algunos resultados.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Guillermo Cortiñas (Universidad de Buenos Aires, Argentina) - titulo: Algebraic K theory of 3-manifold groups abstract: |We describe the algebraic K-theory groups \(K_i(Z[G[))\) where G is the fundamental group of a closed compact 3-manifold and all \(i\in Z\).
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Daniel Juan (Universidad Nacional Autónoma de México, México) - titulo: Clasificación homotópica de álgebras de Leavitt simples puramente infinitas abstract: |En esta charla hablaremos del problema de clasificación de álgebras de Leavitt simples puramente infinitas sobre un grafo finito \(E\) y un cuerpo \(\ell\). Dado un grafo \(E\) tenemos asociado una \(\ell\)-álgebra de Leavitt \(L(E)\). Es un problema abierto decidir cuando el par \((K_0(E), [1_{L(E)}])\), que consiste del grupo de Grothendieck junto con la clase \([1_{L(E)}]\) de la identidad es un invariante completo, a menos de isomorfismo, para la clasificación de álgebras de Leavitt simples puramente infintas de grafo finito. Mostraremos que el par \((K_0(E), [1_{L(E)}])\) es un invariante completo para la clasificación a menos de equivalencia homotópica polinomial. Para probar esto vamos a estudiar la \(K\)-teoría álgebraica bivariante entre álgebras de Leavitt.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Diego Montero (Despegar, Argentina) - titulo: The algebraic K-theory of the Hilbert modular group abstract: |In this talk I will describe how to compute the algebraic K-theory of the Hilbert modular group using the Farrell-Jones conjecture. Among the tools we use are the p-chain spectral sequence, models for the classifying space for the family of virtually cyclic subgroups, and the inductive structure of the equivariant homology theory that appears in the statement of the Farrell-Jones conjecture. This is joint work with M. Bustamante and M. Velásquez
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Luis Jorge Sanchez Saldaña (Universidad Nacional Autónoma de México, México) - titulo: A Gillet-Waldhausen Theorem for chain complexes of sets abstract: |In recent work, Campbell and Zakharevich introduced a new type of structure, called ACGW-category. These are double categories satisfying a list of axioms that seek to extract the properties of abelian categories which make them so particularly well-suited for algebraic K-theory. The main appeal of these double categories is that they generalize the structure of exact sequences in abelian categories to non-additive settings such as finite sets and reduced schemes, thus showing how finite sets and schemes behave like the objects of an exact category for the purpose of algebraic K-theory.
In this talk, we will explore the key features of ACGW categories and the intuition behind them. Then, we will move on to ongoing work with Brandon Shapiro where we further develop this program by defining chain complexes and quasi-isomorphisms for finite sets. These satisfy an analogue of the classical Gillet--Waldhausen Theorem, providing an alternate model for the K-theory of finite sets.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Maru Sarazola (Johns Hopkins University, Estados Unidos) - titulo: Homología de Hochschild topológica y métodos de traza abstract: |Para un anillo \(R\), o más en general un espectro en anillos ("ring spectrum"), Quillen construyó la K-teoría algebraica superior de \(R\) como un espectro en anillos \(K(R)\). Se puede construir otro espectro \(THH(R)\), la homología de Hochschild topológica de \(R\): es un análogo de la homología de Hochschild clásica, solo que en vez de ser sobre el anillo de los enteros \(\mathbb{Z}\), es sobre una base más profunda, el espectro en anillos \(S\), llamado "espectro de las esferas" ("sphere spectrum"). La relación está dada por un mapa \(K(R) \rightarrow THH(R)\) llamado traza, o traza de Dennis topológica, que a veces permite acercarse al cálculo de \(K(R)\). Esbozaremos estas construcciones y daremos un ejemplo de cálculo.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Bruno Stonek (Instituto de Matemáticas de la Academia Polaca de Ciencias, Polonia) - titulo: Topología controlada y conjeturas de isomorfismo en \(K\)-teoría abstract: |Sean \(R\) un anillo, \(G\) un grupo, \(\mathcal{F}\) una familia de subgrupos y \(E_{\mathcal{F}}G\) el \(G\)-\(CW\)-complejo universal con isotropía en \(\mathcal{F}\). La conjetura de isomorfismo identifica, mediante un morfismo de ensamble, la \(K\)-teoría del anillo de grupo \(\mathbf{K}(RG)\) con una teoría de homología equivariante evaluada en \(E_{\mathcal{F}}G\). Una construcción de esta teoría de homología se puede hacer utilizando topología controlada. En este contexto la conjetura afirma que el morfismo de ensamble \[\partial_{\mathcal{F}_n}: K_{n+1}(\mathcal{D}^G(E_{\mathcal{F}}G))\to K_n(RG)\] es un isomorfismo. Para \(\mathcal{F}=\mathcal{V}cyc\) la familia de subgrupos virtualmente cíclicos, la conjetura se conoce como \emph{conjetura de Farrell-Jones} y ha sido probada para una gran clase de grupos.
En este trabajo consideramos los casos en que \(G=\langle t\rangle\) es el grupo cíclico infinito y \(\mathcal{F}\) es la familia trivial, y \(G=D_{\infty}\) es el grupo diedral infinito con la familia de subgrupos finitos. En ambos casos \(\mathbb{R}\) es un modelo para \(E_{\mathcal{F}}G\) y su métrica nos permite dar una noción de tamaño a los morfismos de \(RG\)-módulos. Mostraremos c\'omo utilizar técnicas de control para analizar la suryectividad del morfismo de ensamble \(\partial_{\mathcal{F}_1}\). En el caso en que \(G=\langle t\rangle\) el morfismo de ensamble se identifica con el morfismo de Bass-Heller-Swan \[K_0(R)\oplus K_1(R)\to K_1(R[t^{-1},t]),\] que resulta un isomorfismo para R regular. (Trabajo en conjunto con E. Ellis, E. Rodríguez Cirone y S. Vega.)
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Gisela Tartaglia (Universidad Nacional de La Plata, Argentina) - titulo: Proper actions and equivariant K-theory of group extensions abstract: |In this talk we present some decompositions appearing in equivariant K-theory. Let \[0\to A\to G\to Q\to0\] be an extension of Lie groups with \(A\) compact, let \(X\) be proper \(Q\)-space, we present a decomposition of the \(G\)-equivariant K-theory of \(X\) in terms of twisted equivariant K-theories of \(X\) respect to certain subgroups of \(Q\). We present some examples of computations using this decomposition. Finally we give some ideas of how to extend this decomposition in the context of C*-algebras.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Mario Velazquez (Universität Göttingen, Alemania) - sesion: Stochastic analysis and stochastic processes zoom: https://us02web.zoom.us/j/87489452629 organizadores: - nombre: Paavo Salminen mail: paavo.salminen@abo.fi - nombre: Antoine Lejay mail: Antoine.Lejay@univ-lorraine.fr - nombre: Ernesto Mordecki mail: mordecki@cmat.edu.uy charlas: - titulo: Optimal couplings for interacting particle systems via optimal transport abstract: |We will review some ideas recently developed in order to obtain optimal convergence rates for mean-field stochastic particle systems interacting through binary jumps, by constructing trajectorial couplings that make use of optimal transport plans for empirical measures. Examples of applications in kinetic theory and in branching population models will be discussed.
Based on joint works with Roberto Cortez and Felipe Muñoz.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Joaquin Fontbona (Universidad de Chile, Chile) - titulo: Estimating the parameter of the Skew Brownian motion abstract: |The Skew Brownian motion may be constructed by choosing randomly the sign of a Brownian excursion. It thus depends on a parameter \(\theta\in[-1,1]\). In this talk, we discuss the statistical properties of the maximum likelihood estimator (MLE) for this parameter \(\theta\). In particular, it is consistent and asymptotically mixed normal as the local time of the process is involved in the limit, yet with a rate \(1/4\) and not \(1/2\) as usual. We also discuss some non-asymptotic properties of the MLE.
Based on joint work with Sara Mazzonetto (IECL, Université de Lorraine, Nancy).
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Antoine Lejay (Institut Élie Cartan de Lorraine, Francia) - titulo: Diffusion through a two-sided membrane abstract: |We study the Markov chains on \(\mathbb{Z}^d\), \(d\geq 2\), that behave like a standard symmetric random walk outside of the hyperplane (membrane) \(H=\{0\}\times \mathbb{Z}^{d-1}\). The transition probabilities on the membrane \(H\) are periodic and also depend on the incoming direction to \(H\), that makes the membrane \(H\) two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a \(d\)-dimensional diffusion whose first coordinate is a skew Brownian motion and the other \(d-1\) coordinates is a Brownian motion with a singular drift controlled by the local time of the first coordinate at \(0\). In the proof we utilize a novel martingale characterization of the Walsh Brownian motion and determine the effective permeability and slide direction. Eventually, a similar convergence theorem is established for the one-sided membrane without slides and random iid transition probabilities.
Based on joint work with A. Pilipenko (Institute of Mathematics, NAS of Ukraine, Kiev).
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Ilya Pavlyukevich (Universität Jena, Alemania) - titulo: Non-zero-sum optimal stopping game with continuous versus periodic observations abstract: |We introduce a new non-zero-sum game of optimal stopping with asymmetric information. Given a stochastic process modelling the value of an asset, one player has full access to the information and observes the process completely, while the other player can access it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a Lévy process with positive jumps, we not only prove the existence, but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function.
Based on joint work with Kazutoshi Yamazaki and Neofytos Rodosthenous.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: José Luis Pérez Garmendia (Centro de Investigación en Matemáticas, México) - titulo: 'Diffusion spiders: resolvents, excessive functions and optimal stopping' abstract: |Let \(\Gamma\) be a graph with one internal node \(\mathbf{0}\) and \(n\geq 1\) edges meeting each other at \(\mathbf{0}\). For \(x>0\) and \(i\in\{1,2,\dots,n\}\) we let \(\mathbf{x}=(x,i)\) denote the point on \(\Gamma\) located on the \(i\)'th edge at the distance \(x\) from \(\mathbf{0}\).
Let \(X=(X_t)_{t\geq 0}\) be a one-dimensional diffusion on \(\mathbf{R}_+\) hitting 0 with probability one. A (homogeneous) diffusion spider \(\mathbf{X}=(\mathbf{X}_t)_{t\geq 0}\) is a continuous strong Markov process on \(\Gamma\) which on each edge before hitting \(\mathbf{0}\) behaves as \(X\) before hitting 0. When reaching \(\mathbf{0}\) the spider chooses, roughly speaking, an edge with some given probability to continue the movement. A rigorous construction of \(\mathbf{X}\) can be done using the excursion theory.
In this talk we give an explicit expression for the resolvent kernel of \(\mathbf{X}\). Using this we study the excessive functions of \(\mathbf{X}\) and derive the so called glueing condition to be satisfied at \(\mathbf{0}\). We apply the representation theory (Martin boundary theory) of excessive functions and calculate the representing measure of a given excessive function.
This machinery can be used to solve optimal stopping problems for \(\mathbf{X}\). As an example we consider Walsh's Brownian spider where the diffusion \(X\) is a Brownian motion. We focus on the case where the reward is continuous at \(\mathbf{0}\), linear on the edges and such that the resulting stopping region is connected.
Based on joint work with Jukka Lempa and Ernesto Mordecki.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Paavo Salminen (Åbo Akademi University, Finlandia) - titulo: On the consistency of the least squares estimator in models sampled at random times driven by long memory noise abstract: |In numerous applications data are observed at random times. Our main purpose is to study a model observed at random times incorporating a long memory noise process with a fractional Brownian Hurst exponent \(H\). In this talk, we propose a least squares (LS) estimator in a linear regression model with long memory noise and a random sampling time. The strong consistency of the estimator is established, with a convergence rate depending on \(N\) and Hurst exponent. A Monte Carlo analysis supports the relevance of the theory and produces additional insights.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Soledad Torres (Universidad de Valparaiso, Chile) - titulo: Sufficient variability of paths and differential equations with BV-coefficients abstract: |Partial differential equations and differential systems play a fundamental role in many aspects of our daily life. However, in many applications, especially in the field of stochastic differential or partial differential equations, the underlying equation does not make sense in a classical way, and one has to consider integral equations instead. Moreover, even the concept of integral is subtle. For example, a typical situation is that one needs to consider \(\int_0^t \varphi(X_s)\,dY_s\), where \(\varphi\) is a given function and \(X,Y\) are some continuous but non-differentiable objects. Several powerful techniques such as rough path theory have emerged to treat these situations, and it can be safely stated that nowadays differential systems driven by rough signals are already rather well understood. The idea in the rough path theory is, roughly speaking, that one assumes additional smoothness on the function \(\varphi\) in order to compensate bad behaviour of signal functions \(X\) and \(Y\). As such, the methodology cannot be applied in any straightforward manner if one allows discontinuities in \(\varphi\). In particular, this is the case if \(\varphi\) is a general BV-function.
In this talk we combine tools from fractional calculus and harmonic analysis to some fine properties of BV-functions and maximal functions, allowing us to give a meaningful definition for (multidimensional) integrals \(\int_0^t \varphi(X_s)\,dY_s\) with a BV-function \(\varphi\), provided that the functions \(X\) and \(Y\) are regular enough in the Hölder sense. Here enough regularity means better Hölder regularity than what is customary assumed in the rough path theory, and this gain in regularity can be used to compensate ill behaviour of \(\varphi\). The key idea is that the signal \(X\) should not spend too much time, in some sense, on the bad regions of \(\varphi\). We quantify this in terms of potential theory and Riesz energies. We also discuss several consequences, and provide existence and uniqueness results for certain differential systems involving BV-coefficients. Extensions and further topics are discussed.
Based on joint work with Michael Hinz (Bielefeld University) and Jonas Tölle (University of Helsinki).
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Lauri Viitasaari (Aalto University, Finlandia) - sesion: Control and Stabilization for Partial Differential Equations zoom: https://us02web.zoom.us/j/86141088290 organizadores: - nombre: Fágner D. Araruna mail: fagner@mat.ufpb.br - nombre: Eduardo Cerpa mail: eduardo.cerpa@usm.cl - nombre: Luz de Teresa mail: deteresa@matem.unam.mx charlas: - titulo: Small-time global exact controllability to the trajectories of the Boussinesq system abstract: |In this talk, we consider the global exact controllability problem to the trajectories of the Boussinesq system. We show that it is possible to drive the solution to the prescribed trajectory in small time by acting on the system through the velocity and the temperature on an arbitrary small part of the boundary. The proof relies on three main arguments. First, we transform the problem into a distributed controllability problem by using a domain extension procedure. Then, we prove a global approximate controllability result by following a strategy of Coron and collaborators, which deals with the Navier-Stokes equations. This part relies on the controllability of the inviscid Boussinesq system and asymptotic boundary layer expansions. Finally, we conclude with a local controllability result that we establish with the help of a linearization argument and appropriate Carleman estimates.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Felipe Chaves (Universidade Federal de Paraiba, Brasil) - titulo: 'Identification of a boundary obstacle in a Stokes fluid with Navier–slip boundary conditions: an exterior approach' abstract: |The problem of identifying an obstruction into a fluid duct has several major applications, for example in medicine the presence of a stenosis in a coronary vessels is a life threaten disease. In this talk we formulate a continuous setting and study from a numerical perspective the inverse problem of identifying an obstruction contained in a 2D elastic duct where a Stokes flow becomes turbulent after hitting the boundary (Navier--slip boundary conditions), generating an acoustic waves. To be precise, by using acoustic wave measurements at certain points at the exterior to the duct, we are able to identify the location; extension and height of the obstruction. Thus, our framework constitutes an external approach for solving this obstacle inverse problem. Synthetic examples are used in order to verify the effectiveness of the proposed numerical formulation.
In collaboration with: L. Breton, J. López Estrada and C. Montoya
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Pedro González Casanova (Universidad Nacional Autónoma de México, México) - titulo: Controllability from the exterior of fractional heat equation abstract: |In this talk, we consider the controllability problem from the exterior for the one dimensional heat equation on the interval \((-1,1)\) associated with the fractional Laplace operator \((-\partial_x^2)^s\), where \(0\lt s\lt1\).
In the first part, we will show that there is a control function which is localized in a nonempty open set \(\mathcal O\subset \left(\mathbb{R}\setminus(-1,1)\right)\), that is, at the exterior of the interval \((-1,1)\), such that the system is null controllable at any time \(T \gt 0\) if and only if \(\frac 12 \lt s \lt1\).
Then, in the second part, we study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator on the interval \((-1,1)\). Our control function is localized in an open set \(\mathcal O\) in the exterior of \((-1,1)\). We will show that there exists a minimal (strictly positive) time \(T_{\rm min}\) such that the fractional heat dynamics can be controlled from any initial datum in \(L^2(-1,1)\) to a positive trajectory through the action of an exterior positive control, if and only if \(\frac 12 \lt s \lt 1\). In addition, we prove that at this minimal controllability time, the constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. Finally, we provide several numerical illustrations that confirm our theoretical results.
start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Sebastián Zamorano Aliaga (Universidad de Santaigo de Chile, Chile) - titulo: On the controllability of a model system for long waves in nonlinear dispersive media abstract: |Considered here is a higher order generalization of the classical Boussinesq system which models the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal or the propagation of long-crested waves on large lakes and oceans. Our aim is to investigate the controllability properties of this nonlinear model in terms of the values of the parameters involved in the system. We give general conditions which ensure both the well-posedness and the local exact controllability of the nonlinear problem in some well chosen Sobolev spaces.
Joint work with G. J. Bautista Sanchez (Universidad Privada del Norte, Peru) and S. Micu (University of Craiova, Romania).
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Ademir Pazoto (Universidade Federal do Rio de Janeiro, Brasil) - titulo: 'Nematic liquid crystals: well posedness, optical solitons and control' abstract: |In this talk we present results on well-posedness, decay, soliton solutions and control of the coupled nonlinear Schr\"odinger (NLS) equation \[ \begin{align*} & \partial_{z}u= \frac{1}{2} \mathrm{i} \nabla^{2} u+ \mathrm{i} \gamma (\sin^2(\psi+\theta_0)-\sin^2(\theta_0)) u,\\ &\nu \nabla^{2} \psi= \frac{1}{2}E_0^2\sin(2\theta_0)-\frac{1}{2}(E_0^2+|u|^{2})\sin(2(\psi+\theta_0)), \end{align*} \] where \(u\) and \(\psi\) depend on the ``optical axis'' coordinate \(z \in \mathbb{R}\), and the ``transverse coordinates'' \((x, y) \in \mathbb{R}^2\). Also \(\nabla^{2} = \partial_x^2 + \partial_y^2\) is the Laplacian in the transverse directions, \(E_0\), \(\nu\) and \(\gamma \) are positive constants, and \(\theta_0\) is a constant satisfying \(\theta_0 \in (\pi/4, \pi/2)\). The model arises in the study of optical beam propagation in nematic liquid crystals, and in particular a set of experiments by Assanto and collaborators since 2000. The complex field \(u\) represents the electric field amplitude of a linearly polarized laser beam that propagates through a nematic liquid crystal along the optical axis \(z\). The elliptic equation describes the effects of the beam electric field on the local orientation(director field) of the nematic liquid crystal and has an important regularizing effect, seen experimentally and understood theoretically in related models. The ``director field'' \(\psi + \theta_0\) is a field of angles that describes the macroscopic orientation of the nematic liquid crystal molecules. The laser beam causes an additional deviation \(\psi\) in the orientation of the liquid crystal molecules.
In this talk we will show well posedness of the coupled system, existence of stationary solutions and a ``saturation'' effect consistent with a bound \( \psi+\theta_0 < \pi/2 \) on the total angle. Finally, we will discuss some recent results concerning an optimal control problem where the external electric field varying in time is the control.
Work in collaboration with: Juan Pablo Borgna, Panayotis Panayotaros, Diego Rial.
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Constanza Sánchez de la Vega (Universidad de Buenos Aires, Argentina) - titulo: Stabilization Aspects of the Boussinesq System on a Bounded Interval abstract: |In this talk, we present some recent results related to the rapid boundary stabilization for the Boussinesq System of the KdV-KdV Type on a Bounded interval introduced by J. Bona, M. Chen and J.-C. Saut: $$ \left\{ \begin{array} [c]{l}% \eta_{t}+v_{x}+( \eta v) _{x}+av_{xxx}-b\eta_{xxt}=0\text{,}\\ v_{t}+\eta_{x}+vv_{x}+c\eta_{xxx}-dv_{xxt}=0. \end{array} \right. $$
This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet--Neumann boundary controls acting at the right end point of the interval. Firstly, we build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the nonlinear closed-loop system decays exponentially to zero in the \(L^2(0,L)\)--norm and the decay rate can be tuned to be as large as desired if the initial data is small enough under the effects of two boundary feedback. Moreover, by using a Gramian-based method introduced by Urquiza to design our feedback control, we show that the solutions of the linearized system decay uniformly to zero when the feedback control is applied. The decay rate can be chosen as large as we want. The main novelty of our work is that we can exponentially stabilize this system of two coupled equations using only one scalar input.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Fernando A. Gallego (Universidad Nacional de Colombia, Colombia) - titulo: A unified strategy for observability of waves with several boundary conditions abstract: |We study the controllability of the wave equation acting on an annulus \(\Omega \subset \mathbb{R}^2\), i.e. \( \Omega = B_{R_1} \setminus B_{R_0}\), where \( 0 \lt R_0 \lt R_1\) and \(B_R\) denotes the ball in \(\mathbb{R}^2\) with center at the origin and radius \(R >0\). We are interested in the case of a single control \(h\) acting on the exterior part of the boundary, with a given boundary condition imposed on the interior boundary: $$ \left\{ \begin{array}{ll} \partial_{tt} y - \Delta y = 0, & \hbox{ in } (0,T) \times \Omega, \\ {\mathcal B}y = 0, & (0,T) \times \partial B_{R_0}. \\ y(t,x) = {h(t,x)}, & \hbox{ on } (0,T) \times \partial B_{R_1}. \end{array} \right. $$
We provide a robust strategy to prove the observability of this system with any of several boundary conditions on the internal boundary \((0,T) \times \partial B_{R_0}\), as Fourier conditions, dynamic conditions, conditions with the fractional Laplacian or fluid-structure models.
The main tool we use is given by resolvent characterizations, and the use of adequate estimates. We prove observability estimates with observations performed on the whole external boundary, which are valid for all the mentioned cases for boundary conditions at \(\partial B_{R_0}\). We will mention also recent related results concerning wave and Schrodinger equations in more general domains with the same topological structure.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Alberto Mercado (Universidad Técnica Federico Santa María, Chile) - titulo: Internal controllability of the Kadomtsev-Petviashvili II equatio abstract: |In this talk, we present the internal control problem for the Kadomstev-Petviashvili II equation, better known as KP-II. The problem is studied first when the equation is set in a vertical strip proving by the Hilbert Unique Method and semiclassical techniques proving the internal controllability and second, in a horizontal strip where the controllability in \(L^2(\mathbb{T})\) cannot be reached.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Ivonne Rivas (Universidad del Valle, Colombia), joint with Chenmin Sun - sesion: Geometría Diferencial zoom: https://us02web.zoom.us/j/87314362272 organizadores: - nombre: Romina Arroyo mail: romina.melisa.arroyo@unc.edu.ar - nombre: Viviana del Barco mail: delbarc@ime.unicamp.br - nombre: Silvio Reggiani mail: reggiani@fceia.unr.edu.ar charlas: - titulo: Pluriclosed metrics and Kähler-like conditions on complex manifolds abstract: |A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) or pluriclosed if the torsion of the associated Bismut connection associated is closed. I will present some general results on pluriclosed metrics in relation to symplectic geometry, the pluriclosed flow and Kähler-like curvature conditions.
start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Anna Fino (Università di Torino, Italia) - titulo: The prescribed cross curvature problem abstract: |Chow and Hamilton introduced the notion of cross curvature and the associated geometric flow in 2004. Several authors have built on their work to study the uniformisation of negatively curved manifolds, Dehn fillings, and other topics. Hamilton conjectured that it is always possible to find a metric with given positive cross curvature on the three-sphere and that such a metric is unique. We will discuss several results that support the existence portion of this conjecture. Next, we will produce a counterexample showing that uniqueness fails in general.
start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Artem Pulemotov (The University of Queensland, Australia), joint with Timothy Buttsworth (The University of Queensland) - titulo: Soliton solutions to the curve shortening flow on the 2-dimensional hyperbolic space abstract: |We prove that a curve is a soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space if and only if its geodesic curvature is given as the inner product between its tangent vector field and a vector of the 3-dimensional Minkowski space. We prove that there are three classes of such solutions and for each fixed vector there exits a 2-parameter family of soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space. Moreover, we prove that each soliton is defined on the whole real line, it is embedded and its geodesic curvature, at each end, converges to a constant
start: 2021-09-14T15:00 end: 2021-09-13T15:45 speaker: Keti Tenenblat (Universidade de Brasília, Brasil), joint with Fabio Nunes da Silva (Universidade Federal do Oeste da Bahia) - titulo: Upper bound on the revised first Betti number and torus stability for RCD spaces abstract: |Gromov and Gallot showed that for a fixed dimension \(n\) there exists a number \(\varepsilon(n)>0\) so that any \(n\)-dimensional riemannian manifold \((M,g)\) satisfying \(\textrm{Ric}_g \textrm{diam}(M,g)^2 \geq -\varepsilon(n)\) has first Betti number smaller than or equal to \(n\). In the equality case, \(\textrm{b}_1(M)=n\), Cheeger and Colding showed that then \(M\) has to be bi-Holder homeomorphic to a flat torus. This part can be seen as a stability statement to the rigidity result proven by Bochner, namely, closed riemannian manifolds with nonnegative Ricci curvature and first Betti number equal to their dimension have to be a torus.
The proofs of Gromov and, Cheeger and Colding rely on finding an appropriate subgroup of the abelianized fundamental group to pass to a nice covering space of \(M\) and then study the geometry of the covering. In this talk we will generalize these results to the case of \(RCD(K,N)\) spaces, which is the synthetic notion of riemannian manifolds satisfying \(\text{Ric} \geq K\) and \(\text{dim} \leq N\). This class of spaces include Ricci limit spaces and Alexandrov spaces.
start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Raquel Perales (Universidad Nacional Autónoma de México, México), joint with Ilaria Mondello (Université de Paris Est Créteil) and Andrea Mondino (Oxford University) - titulo: Left-invariant metrics on six-dimensional nilpotent Lie groups abstract: |In this talk we determine the moduli space, up to isometric automorphism, of left-invariant metrics on a family of \(6\)-dimensional Lie group \(H\). We also investigate which of these metrics are Hermitian and classify the corresponding complex structures. This talk is based on a joint work with Silvio Reggiani, based on ``The moduli space of left-invariant metrics of a class of six-dimensional nilpotent Lie groups'', arXiv:2011.02854
start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Francisco Vittone (Universidad Nacional de Rosario, Argentina), joint with Silvio Reggiani (Universidad Nacional de Rosario) - titulo: On the structure of homogeneous Riemannian manifolds with nullity abstract: |We will speak about a joint project, with {\it Antonio J. Di Scala} and {\it Francisco Vittone}, for the study of the structure of irreducible homogeneous Riemannian manifolds \(M^n= G/H\) whose curvature tensor has a non-trivial nullity. In a recent paper we developed a general theory to deal with such spaces. By making use of this theory we were able to construct the first non-trivial examples in any dimension. The key fact is the existence of a non-trivial transvection \(X\) at \(p\) (i.e.\((\nabla X)_p = 0\)) such that \(X_p\notin \nu_p\) (the nullity subspace at \(p\)), but the Jacobi operator of \(X_p\) is zero. The nullity distribution \(\nu\) is highly non-homogeneus in the sense that no non-trivial Killing field lie in \(\nu\) (an so \(\nu\) is not given by the orbits of an isometry subgroup of \(G\)). One has that the Lie algebra \(\mathfrak g\) of \(G\) is never reductive. The co-nullity index \(k\) must be always at least \(3\) and if \(k=3\), then \(G\) must be solvable and \(H\) trivial. The leaves of the nullity foliation \(\mathcal F\) are closed and isometric to a Euclidean space. Moreover, the stabilizer of a given leaf acts effectively on the quotient space \(M/\mathcal F\) and so it does not admit a Riemannian \(G\)-invariant metric. Our main new result is that the so-called {\it adapted} tranvections lie in an abelian ideal \(\mathfrak a\) of \(\mathfrak g\). The distribution \(q\mapsto \mathfrak a .q + \nu _q\) is integrable and does not depend on the presentation group \(G\) and so it defines a geometric foliation \(\hat {\mathcal F}\) on \(M\) (by taking the clausure of the leaves). Moreover, the nullity \(\nu\) is parallel along any leaf of \(\hat {\mathcal F}\) and the projection to the quotient space \(M/ \hat {\mathcal F}\) is a Riemannian submersion. We intend to relate the geometry of \(M\) to that of this quotient.
start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Carlos E. Olmos (Universidad Nacional de Córdoba, Argentina) - titulo: Hermitian and Complex Geometry in Flag Manifolds abstract: |The subject of this talk is a series of results on Hermitian, Complex and Symplectic geometries on the flag manifolds of noncompact semi-simple Lie groups (real or complex). Equations of Differential Geometry in coset spaces of Lie groups are reduced to algebraic equations by homogeneity. Thus many questions are stated and solved in the realm of algebraic properties of the isotropy representation at the tangent space at the origin. In case of the flag manifolds these representations are usually dealt with a combinatorics involving the algebraic structure of the root systems and their Weyl groups.
In this talk some questions and results in this direction will be worked out. Best results are obtained for flag manifolds of the complex groups because the isotropy representations are better behaved. Results on real flag manifolds are more sparse and technically harder. In the complex case it will be presented a more detailed account on the \((1,2)\)-symplectic Hermitian metrics on the maximal (full) flag manifolds.
start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Luiz San Martin (Universidade Estadual de Campinas, Brasil) - titulo: Stability of Einstein metrics on flag manifolds with \(b_2(M)=1\) abstract: |Let \(M\) be a compact differentiable manifold and let \(\mathcal{M}\) denote the space of all unit volume Riemannian metrics on \(M\). Back in 1915, Hilbert proved that the critical points of the simplest curvature functional, given by the total scalar curvature \(Sc:\mathcal{M}_1\rightarrow {\Bbb R}\), are precisely Einstein metrics.
In this talk, after some general preliminaries, we will focus on the case when the metrics and the variations are considered to be \(G\)-invariant for some compact Lie group \(G\) acting transitively on \(M\). As an application, we will give the stability and critical point types of all Einstein metrics on flag manifolds with \(b_2(M)=1\).
start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Jorge Lauret (Universidad Nacional de Córdoba, Argentina) - sesion: Comunicación de las matemáticas en América Latina zoom: https://us02web.zoom.us/j/86796730566 organizadores: - nombre: Javier Elizondo mail: javier@im.unam.mx - nombre: Eduardo Sáenz de Cabezón mail: eduardo.saenz-de-cabezon@unirioja.es charlas: - titulo: Círculos Matemáticos, por el fomento al razonamiento propio... abstract: |Desde los inicios de nuestra educación, la básica y en el hogar, se tiene una marcada tendencia a sancionar el error; además, producto de nuestros tiempos, se tiene la creencia de que la velocidad es un atributo per se. Esta combinación, en la que se sanciona el error y se induce a aprender las cosas a velocidad, lleva a los infantes a recurrir a la memorización indiscriminada; ello paulatinamente alimenta la inseguridad, la fragilidad y a la larga, la inmovilidad.
Es importante entender que cada individuo requiere de tiempos distintos para hacer suya una idea y desarrollarla, y entender también que errar es parte sustancial del aprendizaje. Comprender las cosas a profundidad no sólo favorece la seguridad del ser humano, sino que fomenta la existencia y desarrollo de perspectivas distintas.
Ante esta situación, el objetivo principal del proyecto de Círculos Matemáticos del Instituto de Matemáticas es despertar e infundir en los jóvenes la confianza en su propio razonamiento, así como el respeto de sus propios tiempos.
Desplazar la noción de éxito basada en la velocidad y las actitudes competitivas, y comprender que el error es algo inherente al proceso de aprendizaje, hace que los estudiantes recuperen la confianza en sí mismos y se atrevan a proponer y explorar caminos. Esto concierne no sólo a las matemáticas sino a la vida misma.
start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Laura Ortíz (Universidad Nacional de México, México) - titulo: El abrazo del escutoide abstract: |Sin duda una de las razones más poderosas para insistir en la comunicación de las matemáticas la público general es, al menos para mí, la urgente necesidad de popularización de estas para, sobre todo, acabar con los prejuicios que, tradicionalmente, se han tenido sobre ellas y que conducen, inexorablemente, a la ansiedad de los estudiantes desde las primeras etapas de la educación. Pero, dejando a un lado esta labor del investigador para la sociedad, la divulgación de las matemáticas también pone el trabajo de los investigadores en un escaparate para investigadores de otros ámbitos y puede conducir a la colaboración interdisciplinar. En esta charla les hablaré de cómo la divulgación de la geometría me llevó, nos llevó, a descubrir una nueva forma geométrica mirando a las glándulas salivares de la mosca de la fruta.
start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Clara I. Grima (Universidad de Sevilla, España) - titulo: 'Festival de Matemáticas: la experiencia chilena' abstract: |El Festival de Matemáticas de Chile nació a fines del 2016 como un evento satélite al "Primer Encuentro Conjunto de Matemáticas en Chile y Argentina". Es administrado directamente por la Sociedad de Matemática de Chile y se ha presentado en nueve localidades diferentes del país. En 2020 y 2021 tuvo dos versiones en línea debido a la pandemia COVID19. Se entregará un reporte de la organización de estas actividades, del proyecto de elaboración de material didáctico para las escuelas, y de los planes a futuro de la actividad.
start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Andrés Navas (Universidad de Santiago de Chile, Chile) - titulo: La Comisión de Visibilidad de la Unión Matemática Argentina abstract: |A fines del 2015 se creo la Comisión de Visibilidad de la Unión Matemática Argentina, conmigo a cargo hasta fines del 2019. En esta charla les contare lo que esta comisión intento/intenta realizar y mi lectura de los resultados obtenidos. Se tratará más de un intercambio de ideas que de una charla en sí.
start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Teresa Krick (Universidad de Buenos Aires, Argentina) - titulo: La divulgación matemática como un instrumento de integración social abstract: |Todas las sociedades están formadas por tradiciones, cultura y sobre todo fiestas que crean comunidad y más aún fortalecen el tejido social. De muchas maneras, pareciera que las matemáticas han quedado al margen de estas tradiciones y de la cultura de la sociedades, incluso en muchos casos se han estigmatizado, generando un rechazo per se.
En esta charla platicaré de la importancia de integrar las matemáticas en las tradiciones, en la cultura, y de algunas estrategias que se han desarrollado en el estado de Oaxaca con este fin, a través de las Olimpiadas de matemáticas y el Programa Oaxaqueño de Fortalecimiento a la Educación (PROFE).
start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Bruno Aaron Cisneros de la Cruz (Universidad Nacional Autónoma de México, México) - titulo: Matemáticas (no solo) en las redes abstract: |En los últimos tiempos hemos sido testigos de nuevas formas de divulgación de las matemáticas. Nuevas en cuanto a los formatos y (tal vez) en cuanto a los públicos. En esta charla hablaré sobre una experiencia particular en este ámbito, tanto en España como en Latinoamérica.
start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Eduardo Sáenz De Cabezón Irigaray (Universidad de la Rioja, España) - titulo: Haciendo comunidad en tiempos de confinamiento abstract: |En medio de la incertidumbre de la pandemia de COVID-19 y en el boom de la virtualidad, se han abierto puertas de comunicación entre divulgadores de matemáticas hispanoparlantes en América Latina y en España. En este camino se han organizado encuentros directamente centrados en crear puentes sólidos de colaboración entre distintos grupos.
Durante esta conferencia, presentaré el trabajo que hemos hecho en este sentido Ágata Timón (Instituto de Ciencias Matemáticas), Andrés Navas (Universidad de Chile), Aubin Arroyo (UNAM), Beatriz Vargas (UNAM) y yo. También me gustaría plantear y oír propuestas de caminos para fortalecer la colaboración entre quienes trabajamos divulgando las matemáticas en español.
start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Lucía López de Medrano (Universidad Nacional Autónoma de México, México) - titulo: Narrativa de ficción con temas matemáticos abstract: |En esta charla trataré de abrir la discusión sobre qué significa la narrativa de ficción dentro de la divulgación de las matemáticas.
En este tipo de narrativa se desarrolla una historia en la que se incluye contenido matemático. Surgen varias preguntas: ¿la historia tiene más importancia que el contenido matemático?, ¿se debe de ser muy riguroso con el contenido matemático?
Estas preguntas están relacionadas con la postura de Humberto Eco, quien en un Postcript a su novela "El nombre de la rosa" criticó fuertemente lo que se conoce como Salgarismo: suspender una narración dentro de la trama para realizar una (extensa) explicación científica.
No se busca dar una respuesta a estas preguntas sino plantear algunas posturas y preguntas que permitan avanzar en una discusión sobre este tema.
start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Javier Elizondo (Universidad Nacional Autónoma de México, México) - titulo: La divulgación de las matemáticas en Costa Rica abstract: |Se compartirá la experiencia generada por un grupo de personas que nos hemos dedicado a la divulgación de las matemáticas, con el apoyo de organizaciones no gubernamentales, empresa privada y universidades estatales. En este esfuerzo se han realizado diversas actividades que han tenido un impacto sostenido e interesante. En primer lugar, la producción de “Matemática por un minuto”, una serie de programas para la radio con duración de un minuto cada uno que se transmitieron a nivel nacional dos veces al día y su evolución inesperada hacia videos y posteriormente hacia un libro titulado “Las matemáticas de lo cotidiano”, en sus versiones en español e inglés, así como su edición física y e-book.
Además, el diseño, construcción, experiencias, evolución y clonación del “Museo viajante de ciencias y matemática (MUCYM)” y su impacto en la divulgación de las matemáticas a lo largo y ancho de nuestro país, sobre todo a sectores menos favorecidos de su población.
Por último, la organización de congresos académicos, como el evento bianual “Festival Internacional de Matemáticas” en que se inició en 1998 En las doce ediciones que se han realizado, se amalgamaron mágicamente propuestas didácticas, talleres, ponencias y conferencias con juegos, poesía, museos y teatro entre otros que motivan una participación de profesores, estudiantes y varios grupos sociales que propician el aprendizaje activo de esta bella ciencia, las matemáticas.
start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Manuel Murillo Tsijli (Instituto Tecnológico de Costa Rica, Costa Rica) - titulo: 'FUNDAPROMAT: Lecciones y Logros' abstract: |La Fundación Panameña para la Promoción de las Matemáticas (FUNDAPROMAT) nació el 6 de diciembre de 2019 con el propósito de promover el estudio de las matemáticas en la República de Panamá. Por la pandemia mundial, tuvimos que volvernos creativos para seguir cumpliendo con nuestra misión y comenzamos a organizar eventos virtuales, que son gratis y abiertos a todo público. A la fecha esta Fundación privada sin fines de lucro ha realizado más de 300 eventos virtuales con más de 35,000 participantes, incluyendo panameños y extranjeros de todas partes del mundo. Niños, jóvenes y adultos de todas las edades participan en nuestras actividades. En esta charla, presentaré los diferentes tipos de eventos virtuales que organizamos y compartiré las lecciones aprendidas y los logros alcanzados por la Fundación.
start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Jeanette Shakalli (Fundación Panameña para la Promoción de las Matemáticas, Panamá) - titulo: ¿Qué pasos dar para crear una coordinación hispanoamericana de la divulgación de las matemáticas? abstract: | TBA start: 2021-09-17T16:45 end: 2021-09-17T18:15 speaker: Mesa redonda