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agrega <p> en abstracts completa hasta sesion 35 inclusive

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@ -358,44 +358,44 @@
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.
abstract: <p>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.</p>
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.
abstract: <p>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.</p>
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.
abstract: <p>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.</p>
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.
abstract: <p>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.</p>
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.
abstract: <p>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.</p>
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.
abstract: <p>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.</p>
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.
abstract: <p>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.</p>
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.
<p>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 \(\varphi(x)= \varphi(\tau(x))\).</p>
<p>In this work together with Barbaresco-de Mattos-dos Santos-da Silva, we are considering the same problem for free \(Z_p\) action.</p>
start: 2021-09-15T15:45
end: 2021-09-15T16:30
speaker: Alice Kimie Miwa Libardi (Universidad Estatal Paulista, Brasil)
@ -794,27 +794,27 @@
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
<p>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 <em>conorm code,</em> 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.
where \(x(P_i)\) denotes the residue class of \(x\) modulo \(P_i\) for \(i=1,\ldots,n\).</p>
<p>In this talk the concept of the <em>conorm code,</em> 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.</p>
start: 2021-09-13T15:45
end: 2021-09-13T16:30
speaker: María Chara (Universidad Nacional del Litoral, Argentina)
- titulo: Códigos Reticulados
abstract: |
Reticulados são subconjuntos discretos do espaço euclidiano n dimensional gerados por combinações inteiras de um conjunto de vetores independentes. Reticulados vem sendo usados em processos de codificação para transmissão de sinais particularmente para comunicações do tipo MIMO (multiple-input/multiple-output) e também em esquemas criptográficos na recente área da chamada criptografia pós-quântica. Nesta apresentação faremos uma abordagem geral deste tema e apresentaremos de forma resumida alguns resultados recentes sobre códigos reticulados perfeitos em diferentes métricas, constelações de Voronoi em reticulados construídos a partir de códigos em anéis finitos e uma extensão do problema ``Ring-LWE'' de criptografia baseada em reticulados que inclui uma classe mais geral de reticulados algébricos.
<p>Reticulados são subconjuntos discretos do espaço euclidiano n dimensional gerados por combinações inteiras de um conjunto de vetores independentes. Reticulados vem sendo usados em processos de codificação para transmissão de sinais particularmente para comunicações do tipo MIMO (multiple-input/multiple-output) e também em esquemas criptográficos na recente área da chamada criptografia pós-quântica. Nesta apresentação faremos uma abordagem geral deste tema e apresentaremos de forma resumida alguns resultados recentes sobre códigos reticulados perfeitos em diferentes métricas, constelações de Voronoi em reticulados construídos a partir de códigos em anéis finitos e uma extensão do problema ``Ring-LWE'' de criptografia baseada em reticulados que inclui uma classe mais geral de reticulados algébricos.</p>
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 <em>generalized covering-radius hierarchy</em> 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).
<p>The <em>generalized covering-radius hierarchy</em> 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.</p>
<p>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.</p>
<p>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.</p>
<p>Joint work with Dor Elimelech and Moshe Schwartz (Ben-Gurion University/Israel).</p>
start: 2021-09-14T16:45
end: 2021-09-14T17:30
speaker: Marcelo Firer (Universidade Estadual de Campinas, Brasil)
@ -829,25 +829,25 @@
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.
<p>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.</p>
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.
<p>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.</p>
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.
<p>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.</p>
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).
<p>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).</p>
start: 2021-09-13T16:45
end: 2021-09-13T17:30
speaker: Cintya Wink de Oliveira Benedito (Universidade Estadual Paulista, Brasil)
@ -1090,54 +1090,54 @@
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.
<p>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.</p>
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.
<p>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.</p>
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.
<p>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.</p>
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.
<p>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.</p>
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.
<p>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.</p>
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):
<p>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
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.
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.</p>
<p>Some of the results presented are joint with C. Vasquez, F. Valenzuela, J. Yang and P. Carrasco.</p>
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.
<p>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.</p>
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).
<p>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).</p>
start: 2021-09-16T17:30
end: 2021-09-16T18:15
speaker: Ferrán Valdez (Universidad Nacional Autónoma de México, México)
@ -1214,53 +1214,53 @@
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 <i> symmetric real semigroups</i>.
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.
<p>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 <i> symmetric real semigroups</i>.</p>
<p>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. <br />
II) There exists an infinite symmetric real semigroup (in fact, a fan) which is not realizable by any ring of formal power series. </p>
start: 2021-09-13T15:45
end: 2021-09-13T16:30
speaker: Alejandro Petrovich (Universidad de Buenos Aires, Argentina) joint with Max Dickmann (Université de Paris y Sorbonne Université, Francia)
- titulo: Boolean Real Semigroups
abstract: |
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.
<p>If \(G\) is a real semigroup (RS), write \(G^\times = \{ x \in G : x^2 = 1\}\) for the group of units in \(G\) and <br />
\(Id(G) = \{ e \in G : e^2 = e \}\) for the distributive lattice of idempotents in \(G\)</p>
<p>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.</p>
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.
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>All results and involved notions will be explained and backed up with examples.</p>
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.
<p>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.</p>
<p>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.</p>
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.
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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)).</p>
<p>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.</p>
<p>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.</p>
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.
<p>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.</p>
<p>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.</p>
<p>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.</p>
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)
@ -1403,50 +1403,50 @@
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.
<p>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.</p>
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.
<p>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.</p>
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.
<p>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.</p>
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.
<p>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.</p>
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.
<p>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.</p>
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.
<p>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.</p>
<p>Joint work with Nathan Ryan, Zachary Scherr and Stephanie Treneer.</p>
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).
<p>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).</p>
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.
<p>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.</p>
start: 2021-09-16T17:30
end: 2021-09-16T18:15
speaker: Ila Varma (University of Toronto)
@ -1678,54 +1678,54 @@
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.
<p>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.</p>
<p>Joint work with A. Fribergh, M. Holmes and E. Perkins.</p>
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.
<p>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).</p>
<p>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 <br />
[2] Franco, T., Gonçalves, P., Neumann, A. (2021): Large Deviations for the SSEP with slow boundary: the non-critical case. Online: arxiv.</p>
start: 2021-09-14T16:45
end: 2021-09-14T17:30
speaker: Adriana Neumann (Universidade Federal do Rio Grande do Sul, Brasil)
- titulo: Procesos de Lévy Libres
abstract: |
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.
<p>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.</p>
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.
<p>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.</p>
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).
<p>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).</p>
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.
<p>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.</p>
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).
<p>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.</p>
<p>Joint work with Gerardo Barrera (University of Helsinki, Finland) and Juan Carlos Pardo (CIMAT, México).</p>
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.
<p>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.</p>
start: 2021-09-14T17:30
end: 2021-09-14T18:15
speaker: Avelio Sepúlveda (Universidad de Chile, Chile)
@ -1993,10 +1993,10 @@
speaker: Walter Ferrer (Universidad de la República, Uruguay)
- titulo: A criterion for an abelian variety to be non-simple and applications
abstract: |
<p>Let \((A,\cL)\) be a complex abelian variety of dimension \(g\) with polarization of type \(D = \diag(d_1,\dots,d_g)\). So \(A = V/\Lambda\) where \(V\) is a complex vector space of dimension \(g\) and \(\Lambda\) is a lattice of maximal rank in \(\CC^g\) 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 \(g\).</p>
<p>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 \((A,\cL)\) is non-simple.</p>
<p>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 \(g\).</p>
<p>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.</p>
<p>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.</p>
<p>In this direction, we use this criterion in combination with the so called {\it 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 \(\QQ[G]\) as a product of minimal (left) ideals which gives an isogeny
<p>In this direction, we use this criterion in combination with the so called <i>Group Algebra Decomposition</i> 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
\[B_1^{n_1} \times \cdots \times B_r^{n_r}\to JX\]</p>
<p>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 \(JX\).</p>
<p>This is a joint work with R. Auffarth and H. Lange.</p>
@ -2075,53 +2075,53 @@
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.
<p>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.</p>
<p>Based on joint works with Roberto Cortez and Felipe Muñoz.</p>
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).
<p>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.</p>
<p>Based on joint work with Sara Mazzonetto (IECL, Université de Lorraine, Nancy).</p>
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).
<p>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.</p>
<p>Based on joint work with A. Pilipenko (Institute of Mathematics, NAS of Ukraine, Kiev).</p>
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.
<p>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.</p>
<p>Based on joint work with Kazutoshi Yamazaki and Neofytos Rodosthenous.</p>
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.
<p>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 \(\mathbf{0}\).</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>Based on joint work with Jukka Lempa and Ernesto Mordecki.</p>
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.
<p>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.</p>
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).
<p>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.</p>
<p>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.</p>
<p>Based on joint work with Michael Hinz (Bielefeld University) and Jonas Tölle (University of Helsinki).</p>
start: 2021-09-16T15:00
end: 2021-09-16T15:45
speaker: Lauri Viitasaari (Aalto University, Finlandia)