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actualizaciones en abstract de sesiones, cambio de horario Lalin <-> Sirolli sesión 35

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mail: brech@ime.usp.br
charlas:
- titulo: Around (*)
abstract: In this talk I will present work motivated by the derivation of the \(\mathbb P_{max}\) axiom \((*)\) from Martin's Maximum\(^{++}\).
abstract: <p>In this talk I will present work motivated by the derivation of the \(\mathbb P_{max}\) axiom \((*)\) from Martin's Maximum\(^{++}\).</p>
start: 2021-09-13T15:00
end: 2021-09-13T15:45
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.
abstract: <p>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.</p>
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.
<p>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.</p>
<p>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.</p>
<p>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.</p>
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.
abstract: <p>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.</p>
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 \(\aleph_1\).
<p>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 \(\aleph_1\).</p>
start: 2021-09-14T16:45
end: 2021-09-14T17:30
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.
<p>Model theory is a branch of mathematical logic with strong interactions with other branches of mathematics, including algebra, geometry and number theory.</p>
<p>In this talk we are interested in differential and difference fields from the model-theoretic point of view.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>In this talk we study groups definable in models of \(DCF_mA\), and show an analogue of Phyllis Cassidy's result.</p>
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.
<p>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.</p>
<p>To achieve this, we worked in the proof assistant Isabelle, basing our development on the theory Isabelle/ZF by L. Paulson and others.</p>
<p>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.</p>
<p>We'll also compare our formalization with the recent one by Jesse M. Han and Floris van Doorn in the proof assistant Lean.</p>
start: 2021-09-13T17:30
end: 2021-09-13T18:15
speaker: 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.
<p>In this talk we present recent developments in the study of non-classical models of ZFC. </p>
<p>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.</p>
<p>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 \(\mathbf{V}^{\mathbb{A}} \vDash \varphi \land \neg \varphi\).</p>
<p>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.</p>
<p>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.</p>
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
@ -99,7 +82,7 @@
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.
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)
@ -148,41 +131,41 @@
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:
end:
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:
end:
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:
end:
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:
end:
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:
end:
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:
end:
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:
end:
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
organizadores:
@ -301,7 +284,7 @@
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: Liliane Basso Barichello (Universidade Federal do Rio Grande do Sul, Brasil)
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.
@ -476,8 +459,8 @@
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 \(\partial_t2\phi -\partial_x2\phi + W'(\phi) = 0, \quad (t,x)\in\R\times\R\). 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.
<p>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.</p>
<p>This is a joint work with Y. Martel, C. Muñoz and H. Van Den Bosch.</p>
start: 2021-09-17T15:45
end: 2021-09-17T16:30
speaker: Michal Kowalczyk (Universidad de Chile, Chile)
@ -513,9 +496,9 @@
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 {\sl 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 {\sl 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 {\sl 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 {\sl 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).
<p>In this talk, we explore combinatorial arrays that are useful in identifying faulty interactions in complex engineered systems. A <i> separating hash family</i> \({\sf SHF}_\lambda(N; k,v,\{w_1,\dots,w_s\})\) 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 <i>separated</i>) 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.</p>
<p>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 <i>covering</i> 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 <i>covering</i> if, for every way to choose \(\{C_1, \dots, C_s\}\), there are at least \(\lambda\) covering rows.</p>
<p>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).</p>
start: 2021-09-17T15:00
end: 2021-09-17T15:45
speaker: Charles J. Colbourn (Arizona State University, Estados Unidos)
@ -662,7 +645,7 @@
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\(\rm {\textbf d}\)
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 \(\mathcal{X}\)
@ -674,7 +657,7 @@
$$\mathcal{X}=\mathcal{O}_B= \{ UBU^* : \, U\quad\text{es unitaria}\}\,.$$
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
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
@ -701,7 +684,9 @@
- 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.
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:
@ -834,10 +819,10 @@
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 <em>grid code</em> \(\codeC\) is a subset of \(G\), if \(\codeC\) is subgroup of \(G\), then it is said that \(\codeC\) is a \texttt{group code}. The elements of \(\codeC\) are called \texttt{codewords}. The \texttt{minimum distance} \(d\) of a code \(\codeC\) is defined as usually, that is, as the smallest distance between any two different elements of \(\codeC\). Let \(\codeC\) be a code of \(G\) with minimum distance \(d\). Then we say that \(\codeC\) is a \((n,|\codeC|,d)\)-code over \(G\) and \((n,|\codeC|,d)\) are its parameters.
In this talk, we consider such codes, and we prove some classical results on block codes, like Singleton Bound and others.
<p>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
$$l_{1}\Big(\prod_{i=1}^n g_i^{\epsilon_i}, \prod_{i=1}^n g_i^{\delta_i}\Big) = \sum_{i=1}^n |\epsilon_i-\delta_i|.$$</p>
<p>A <em>grid code</em> \(\mathscr{C}\) is a subset of \(G\), if \(\mathscr{C}\) is subgroup of \(G\), then it is said that \(\mathscr{C}\) is a <i>group code</i>. The elements of \(\mathscr{C}\) are called <i>codewords</i>. The <i>minimum distance</i> \(d\) 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.</p>
<p>In this talk, we consider such codes, and we prove some classical results on block codes, like Singleton Bound and others.</p>
start: 2021-09-14T17:30
end: 2021-09-14T18:15
speaker: Ismael Gutiérrez (Universidad del Norte, Colombia)
@ -940,7 +925,7 @@
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 \(k>2\) 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-\(\infty\).
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
@ -1145,7 +1130,7 @@
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.
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)
@ -1442,15 +1427,15 @@
- 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: 2021-09-15T16:45
end: 2021-09-15T17:30
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: 2021-09-16T16:45
end: 2021-09-16T17:30
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: |
@ -1519,7 +1504,7 @@
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.
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)
@ -1646,8 +1631,8 @@
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.
<p>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.</p>
<p>Our goal in this talk is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Amp\`ere equation.</p>
Joint work with P. Blanc (Jyväskylä), F. Charro (Detroit), and J.J. Manfredi (Pittsburgh).
start: 2021-09-14T15:00
end: 2021-09-14T15:45
@ -1813,42 +1798,42 @@
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:
end:
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:
end:
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<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:
end:
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:
end:
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:
end:
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:
end:
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
organizadores:
@ -1894,7 +1879,7 @@
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
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.
@ -1959,51 +1944,51 @@
- 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:
end:
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:
end:
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:
end:
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:
end:
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:
end:
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.
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end:
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.
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end:
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.
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end:
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: |
@ -2014,8 +1999,8 @@
\[B_1^{n_1} \times \cdots \times B_r^{n_r}\to JX\]
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.
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end:
start: 2021-09-15T15:00-0300
end: 2021-09-15T15:45-0300
speaker: Anita Rojas (Universidad de Chile, Chile)
- sesion: K-teoría
organizadores:
@ -2208,7 +2193,7 @@
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 \(\Omega \subset \R^2\), i.e. \( \Omega = B_{R_1} \setminus B_{R_0}\), where \( 0 < R_0 < R_1\) and \(B_R\) denotes the ball in \(\R^2\) 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:
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}
@ -2226,7 +2211,7 @@
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 $L^2(\T)$ cannot be reached.
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 \(L^2(\T)\) cannot be reached.
start: 2021-09-14T16:45
end: 2021-09-14T17:30
speaker: Ivonne Rivas (Universidad del Valle, Colombia), joint with Chenmin Sun

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