From 987d64fcf9b01a92343d291054bc2dbdb53726db Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Germ=C3=A1n=20Correa?= Date: Fri, 13 Aug 2021 19:46:31 -0300 Subject: [PATCH] Se agregan todas las sesiones que faltaban --- data/sesiones.yml | 1319 ++++++++++++++++++++++++++++++++++++++++++++- 1 file changed, 1317 insertions(+), 2 deletions(-) diff --git a/data/sesiones.yml b/data/sesiones.yml index f6c9f10..aaf3c6f 100644 --- a/data/sesiones.yml +++ b/data/sesiones.yml @@ -865,6 +865,79 @@ mail: dcarando@dm.uba.ar - nombre: Maite Fenández Unzueta mail: maite@cimat.mx + charlas: + - titulo: Geometría Lipschitz para espacios de operadores + abstract: | + La teoría no lineal de espacios de Banach ha sido un área muy activa en las últimas décadas, pero su contraparte no conmutativa (es decir, la teoría no lineal de espacios de operadores) apenas está siendo explorada. + En un trabajo previo con Bruno M. Braga hemos mostrado que la noción más obvia de transformación Lipschitz entre espacios de operadores desafortunadamente da lugar a una teoría trivial, en el sentido de que las únicas transformaciones que satisfacen la definición son lineales. En este trabajo, conjunto con Bruno M. Braga y Thomas Sinclair, introducimos una noción más sutil de encaje Lipschitz entre espacios de operadores: estrictamente más débil que la noción lineal, pero suficientemente rígida para imponer restricciones en la estructura lineal de los espacios de operadores. Con este propósito introducimos una noción de espacios libres Lipschitz para espacios de operadores, y probamos algunas de sus propiedades al estilo del trabajo clásico de G. Godefroy y N. Kalton. + start: + end: + speaker: Alejandro Chávez Domínguez (University of Oklahoma, Estados Unidos) + - titulo: Singular perturbations of symplectic isomorphism of Banach spaces + abstract: | + Given a real reflexive Banach space \(X\), an onto isomorphism \(\alpha: X \to X^*\) is said to be \emph{symplectic} if \(\alpha^*= - \alpha\) with the canonical identification. We study some properties of strictly singular perturbations of symplectic isomorphism in the direction of the relations obtained by V. Ferenczi and E. Galego (2007) on complex structures of a Banach and its hyperplanes with the elements of square -1 of the algebra \(\mathcal L(X)/S(X)\) quotient of the algebra \(\mathcal L(X)\) of linear and continuous operators on \(X\) by the ideal of strictly singular operators. + start: + end: + speaker: Wilson Cuellar Carrera (Universidade de Sâo Paulo, Brasil) + - titulo: Normas tensoriales en operator spaces + abstract: | + En esta charla presentamos el inicio de una teoría sistemática de normas tensoriales en operator spaces y su interacción con ideales de operadores lineales asociados. Basados en la teoría de productos tensoriales en espacios de Banach, proponemos las definiciones naturales correspondientes a este nuevo ámbito y obtenemos algunos resultados análogos. Sin embargo, hay también notables discrepancias con la teoría clásica que requieren nuevos conocimientos, técnicas, ideas o hipótesis. En particular, mostramos varias diferencias sustanciales y muchas preguntas abiertas en la lista de las denominadas normas naturales (en el sentido de Grothendieck). + Trabajo en colaboración con Alejandro Chávez-Domínguez y Daniel Galicer. + start: + end: + speaker: Verónica Dimant (Universidad de San Andrés, Argentina) + - titulo: Linear dynamics of operators on non-metrizable topological vector spaces + abstract: | + In this talk we will explore the following notions of chaos for operators on non-metrizable topological vector spaces: mixing, hypercyclicity, cyclicity, \(n\)-supercyclicity, Li-Yorke chaos and Devaney chaos. We will discuss the main differences between these notions for operators on Fr\'echet and on non-metrizable topological vector spaces. + To illustrate some of these differences, we will explore results about the linear dynamics of convolution operators on spaces of entire functions of finitely and infinitely many complex variables. A classical result due to Godefroy and Shapiro states that every nontrivial convolution operator on the Fréchet space \(\mathcal{H}(\mathbb{C}^n)\) of all entire functions of \(n\) complex variables is hypercyclic. In sharp contrast to this result, in a joint work with J. Mujica it was showed that no translation operator on the space \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) (which is a complete non-metrizable locally convex space) of entire functions of infinitely many complex variables is hypercyclic. Recently, in a joint work with B. Caraballo it was showed that no convolution operator on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) is cyclic or \(n\)-supercyclic for any positive integer \(n\). In the opposite direction, it was proved that every nontrivial convolution operator on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) is mixing. Exploring the concept of Li-Yorke chaos on non-metrizable topological vector spaces, it was also proved that nontrivial convolution operators on \(\mathcal{H}(\mathbb{C}^\mathbb{N})\) are Li-Yorke chaotic. + start: + end: + speaker: Vinícius V. Fávaro (Universidade Federal de Uberlândia, Brasil) + - titulo: Constante de proyección de espacios de polinomios + abstract: | + Un resultado elemental en la teoría de espacios normados asegura que todo espacio de dimensión finita está complementado en cualquier superespacio que lo contenga. Luego, resulta natural querer cuantificar la norma de una determinada proyección sobre un espacio dado. + Una cantidad clásica relacionada con este propósito es la denominada \emph{constante de proyección} del espacio \( X \), definida como la mejor constante \(c>0\) que asegura para \emph{cualquier} superespacio \( Y \supset X\), la existencia de una proyección de \( Y \) sobre \( X \) de norma menor o igual a \( c \). + En esta charla discutiremos la constante de proyección de \(\mathcal P_m(X_n)\) (el espacio de polinomios \(m\)-homogéneos sobre un espacio \(n\)-dimensional \(X_n\)) y la conexión existente con algunos invariantes geométricos relativos a \(X_n\). Nos focalizamos en la dependencia respecto de ambos parámetros \(m\) y \(n\), i.e., el grado de homogeneidad y la dimensión del espacio, respectivamente. También trataremos la constante de proyección para otras clases de polinomios (e.g., polinomios de Dirichlet, polinomios sobre el cubo Booleano, trigonométricos analíticos, polinomios de grado menor o igual a \(m\) y tetraedrales homogéneos, entre otros). Mencionaremos cómo la constante de proyección de espacios de polinomios se conecta con el estudio de incondicionalidad en espacios de polinomios y el radio de Bohr. + Trabajo en conjunto con A. Defant, M. Mansilla, M. Mastyło y S. Muro. + start: + end: + speaker: Daniel Galicer (Universidad de Buenos Aires, Argentina) + - titulo: Ideals of multilinear operators from a geometric approach + abstract: | + A recent approach for studying collections of multilinear operators, based on the geometry of tensor products, allows to generalize collections of linear operators to the multilinear setting. Besides defining new collections, this approach allows to characterize them in terms of summing properties, factorizations and duality relations with tensor products. The defined collections to this day include compact, summing, dominated and even factorable operators. These collections of multilinear operators enjoy an ideal behavior, that is, they are closed under compositions that preserves the geometry of the multilinear domain. This ideal property, together with the geometry of tensor products, has inspired the study of the so called \(\Sigma\)-ideals. + This talk is dedicated to present the foundations of \(\Sigma\)-ideals and their classification in terms of tensor norms. Moreover we will explore the \(\Sigma\)-ideal of \((p,q)\)-dominated multilinear operators and its classification in terms of tensor spaces as well as the factorization through \(L_p\) spaces that they admit. + start: + end: + speaker: Samuel García Hernández (Universidad Tecnológica de México, México) + - titulo: Geometría de los espacios de sucesiones de Marcinkiewicz + abstract: | + Los espacios de sucesiones de Marcinkiewicz \(m_{\Psi}^0\) y sus duales \((m_\Psi^0)'\) y \(m_\Psi\), son espacios invariantes por reordernamientos, determinados por un símbolo \(\Psi\) (una sucesión no decreciente de números reales positivos). Para símbolos apropiados \(\Psi\), estos espacios devienen en espacios de Lorentz (sus preduales y duales) \(d_*(w,1), d(w,1)\) y \(d^*(w,1)\). + El objetivo de esta charla es entender, para un símbolo general \(\Psi\), la geometría de la bola unidad de \(m_{\Psi}^0,(m_\Psi^0)'\) y \(m_\Psi\) a partir de la caracterización de sus puntos extremales (reales y complejos) y de sus puntos expuestos. + Veremos que los puntos extremales complejos de la bola unidad de \(m_\Psi^0\) están determinados por la geometría de los subespacios finito-dimensionales \(m_\Psi^n\). Mientras que la geometría de la bola unidad de \(m_\Psi\) depende fuertemente del comportamiento asintótico de los valores de \(\Psi\). También caracterizaremos los puntos extremales y expuestos de la bola unidad de \((m^0_\Psi)'\). Como consecuencia, extendemos resultados de Kamińska, Lee y Lewicki (2009) y de Ciesielski y Lewicki (2019). Trabajo en conjunto con Chris Boyd (UCD). + start: + end: + speaker: Silvia Lassalle (Universidad de San Andrés, Argentina) + - titulo: Cluster value problems in Banach spaces + abstract: | + In this talk I will discuss a couple of problems about the Banach algebra \(H^{\infty}(B)\) of bounded holomorphic functions on the ball \(B\) of a Banach space \(X\). First I will talk about the Corona problem, which asks if the evaluations on each point of \(B\) of functions in \(H^{\infty}(B)\) is a dense subset of \(M_{H^{\infty}(B)}\), the nonzero algebra homomorphisms from \(H^{\infty}(B)\) into \(\mathbb{C}\). An easier problem deals with a comparison of the limit behavior of functions in \(H^{\infty}(B)\) towards each point \(x^{**}\) of the closed ball \(\overline{B}^{**}\) of \(X^{**}\) with the elements of \(M_{H^{\infty}(B)}\) that in a sense correspond to \(x^{**}\). I will discuss why the last problem, the cluster value problem, is indeed easier than the Corona problem, and I will talk about some of the essential ideas behind the cluster value theorems known up to now. + start: + end: + speaker: Sofía Ortega Castillo (Universidad de Guadalajara, México) + - titulo: Spaceability and Residuality in some sets of analytic functions on the open unit disk + abstract: | + In this talk we will present some results concerning algebraic and topological structure inside the following sets of special functions: the set \(\mathcal{F}\) of all Bloch functions defined on the open unit disk that are unbounded, and the set \(\mathcal{W}^p\) of all Bergman functions defined on the open unit disk that are not Bloch functions. We show that \(\mathcal{F}\) and \(\mathcal{W}^p\) are spaceable and residual, that is, \(\mathcal{F}\cup\{0\}\) (and \(\mathcal{W}^p\cup\{0\}\)) contains a closed infinite dimensional vector space. Residuality of the set \(\mathcal{F}\) (and \(\mathcal{W}^p\)) means that their complements are of first category. We also investigate the set of all holomorphic functions of bounded type defined on a Banach algebra \(E\) into \(E\), which are not Lorch-analytic. We show that this set is spaceable but not residual. + Joint work with M. Lilian Lourenço, IME-USP, São Paulo, Brazil. + start: + end: + speaker: Daniela M. Vieira ( Universidade de São Paulo, Brasil) + - titulo: Extremos de polinomios - un enfoque probabilístico + abstract: | + Consideremos un polinomio \(k\)-homogéneo \(P:\mathbb R^n \longrightarrow \mathbb R\). ?`Cuál es la probabilidad de que \(P\) alcance un máximo relativo en algún vértice de la bola-1 (i.e., la bola unidad de la norma \(\Vert \cdot \Vert_1\))? ¿Y en un v\'ertice de la bola-\(\infty\)? Se sabe que si \(k>2\) la probabilidad de alcanzar un máximo relativo en algún vértice de la bola-1 tiende a uno a medida que la dimensión \(n\) crece. Esto es falso para \(k=2\), y es un problema abierto para la bola-\(\infty\). + En esta charla veremos algunas de las herramientas utilizadas para encarar estas cuestiones, y algunas de las dificultades que se presentan. + Veremos también un resultado reciente, obtenido en conjunto con Damián Pinasco y Ezequiel Smucler, para polinomios sobre un simple: si \(k>4\), la probabilidad de que un polinomio \(k\)-homogéneo alcance un máximo relativo en algún vértice del simple \(n\)-dimensional tiende a uno al crecer la dimensión \(n\). Esto requiere un aporte a un viejo problema estadístico: el de las probabilidades ortantes. + start: + end: + speaker: Ignacio Zalduendo (Universidad Torcuato Di Tella, Argentina) - sesion: Ecuaciones diferenciales y estructuras geométricas organizadores: - nombre: John Alexander Arredondo García @@ -875,6 +948,79 @@ mail: mikarm@uniandes.edu.co - nombre: Jesús Muciño Raymundo mail: muciray@matmor.unam.mx + charlas: + - titulo: On the averaging theory for computing periodic orbits + abstract: | + This talk deals with the averaging theory, which is a classical tool allowing us to study periodic solutions of the nonlinear differential systems. + We present some applications of averaging theory. Also, we shall show a system where the averaging method fails to detect periodic orbits. + start: + end: + speaker: Martha Alvarez Ramírez (Universidad Autónoma Metropolitana, México) + - titulo: Integrable models close to slow-fast Hamiltonian systems + abstract: | + Slow-fast Hamiltonian systems are characterized by a separation of the phase space into slow and fast parts typically identified by a small (or slow) parameter. This kind of Hamiltonian system is not integrable, in general; even though in one degree of freedom. In this lecture, we study an integrable model associated with a slow-fast Hamiltonian system of two degrees of freedom. Thinking of the slow parameter as a perturbative one, making suitable symmetry assumptions, and using normal form theory, we show that a slow-fast Hamiltonian system in two degrees of freedom is close to an integrable Hamiltonian model. What we gain with this model is the possibility to associate a family of Lagrangian 2-tori which is almost invariant with respect to the original slow-fast Hamiltonian system. As an important application, this family of almost invariant Lagrangian 2-tori can be used to compute approximations to the spectrum of the quantum model associated with the slow-fast Hamiltonian systems. + start: + end: + speaker: Misael Avendaño-Camacho (Universidad de Sonora, México) + - titulo: The Maximum Principle and solutions to nonlinear elliptic problems + abstract: | + In this talk, we will use a form of the maximum principle and an iteration method to show existence of solutions to the Dirichlet problem: + \[\left\{\begin{array}{l}-\Delta u = \lambda\left(x\right)f\left(u\right)+h\left(x\right)\quad \mbox{in}\quad \Omega,\\u=0 \quad \mbox{on}\quad \partial \Omega,\end{array}\right.\]in bounded and thin unbounded domains. Also, we will show how these methods can be applied to show existence of nontrivial solutions to the Lane-Emden system\[\left\{\begin{array}{l}-\Delta u=v^p,\quad -\Delta v=u^p, \quad \mbox{in}\quad \Omega\\u=v=0, \quad\mbox{on}\quad \partial \Omega. \end{array}\right.\] + This is joint work with Jonatán Torres-Orozco. + start: + end: + speaker: Jean C. Cortissoz (Universidad de los Andes, Colombia) + - titulo: Singularities of the Focal Set of a Line Congruence + abstract: | + A line congruence is a \(2\)-dimensional arrangement of lines in \(3\)-space. It is a classical topic of projective differential geometry and has many relations with binary differential equations. In this talk we give a geometric description of the generic singularities of the focal surface of a line congruence. We show that these singularities occur along the ridge and double eigenvalue curves. A basic tool is the support function associated with an eqüiaffine vector field transversal to a surface in \(\mathbb{R}^3\). + This is a joint work with Ronaldo Garcia (UFG, Brazil). + start: + end: + speaker: Marcos Craizer (Pontifícia Universidade Católica do Rio de Janeiro, Brasil) + - titulo: Line congruence in singular surface in \(\mathbb R^3\) + abstract: | + A line congruence in the Euclidean space of dimension \(3\) is a \(2\) parameter family of lines in \(\mathbb R^{3}\). + The first record about line congruences appeared in "Mémoire sur la Théorie des Déblais et des Remblais" (1776,1784) where Gaspard Monge seeks to solve a minimizing cost problem of transporting an amount of land from one place to another, preserving the volume. + After Monge, Ernst Eduard Kummer in ''Allgemeine Theorie der geradlinigen Strahien systeme", was the first to deal exclusively with the general theory of line congruences. Mainly due to optical applications, line congruences started to gain importance, increasing even more with the development of technology. In this lecture, we will deal with line congruence when the parameters vary in a surfaces with singularities. We will build the theory of congruence lines when the parameter space is a Frontal. As application, we will consider line congruence of cuspidal edges. We will show relations between singularities of the congruence lines and geometric properties of initial cuspidal edges. + This a joint work with Tito Medina(ICMC/USP), Igor Chagas Santos(ICMC/USP) and Maria Aparecida S. Ruas(ICMC/USP). + start: + end: + speaker: Débora Lopes da Silva (Universidade Federal de Sergipe, Brasil) + - titulo: Frobenius manifolds in the context of \(\mathbb{A}\)-manifolds + abstract: | + The theory of Frobenius manifolds has important relevance in mathematics since the application to enumerative geometry with the counting of the number of zero genus curves of degree \(d\) passing through \(3d-1\) points in the projective space \(\mathbb{C} P^2\) (M. Kontsevich). + Let \(\mathbb{A}\) be a finite-dimensional commutative associative algebra with unit over \(\mathbb{R}\). A map \(F: \mathbb{A}^n \to \mathbb{A}^n\) is called \(\mathbb{A}\)-differentiable if \(dF\) is \(\mathbb{A}\)-lineal. Let \(\Gamma_\mathbb{A}\) be the pseudogroup of local \(\mathbb{A}\)-diffeomorphisms of \(\mathbb{A}^n\). An \(\mathbb{A}\)-manifold is a manifold endowed with \(\Gamma_\mathbb{A}\)-atlas. An important example of \(\mathbb{A}\)-manifold is the total space of A. Weil's bundle of \(\mathbb{A}\)-closed points. + The theory of \(\mathbb{A}\)-manifolds (A.P. Shirokov, V.V. Vishnevskii, G.I. Kruchkovich, V.V. Shurygin) has strong relations to the foliation theory and to geometry of jet bundles and the theory of natural operations in differential geometry (I. Kolář, J. Slovák, P. W. Michor). + In our talk we will show that Frobenius manifolds in the sense of B. Dubrovin and N.J. Hitchin have the structure of an \(\mathbb{A}\)-manifold, where \(\mathbb{A}\) is a Frobenius algebra. We will introduce Weil coalgebras, the \(G\)-equivariant graduated versions of Weil algebras, and the corresponding bundle structures. + start: + end: + speaker: Mikhail Malakhaltsev (Universidad de los Andes, Colombia) and Carlos Segovia (Universidad Nacional Autónoma de México, México) + - titulo: Classification of complex polynomials in $\mathbb{C}^2$ and its applications + abstract: | + The huge group of polynomial diffeomorphisms of \(\mathbb{C}^2\) acts on the space of polynomials \(\{ f: \mathbb{C}^2 \longrightarrow \mathbb{C} \}\) by coordinate changes. The classification of polynomials under this action is a challenging open problem. We obtain a very concrete result for degree three polynomials. An application to perturbation of singular Hamiltonian foliations on \(\mathbb{C}^2\), having \(\mathbb{C} \backslash \{ k \, \hbox{points}\}\), as generic leafs, is provided. + Joint work with John Alexander Arredondo (Colombia) and Salomón Rebollo (Chile). + start: + end: + speaker: Jesús R. Muciño-Raymundo (Universidad Nacional Autónoma de México, México) + - titulo: Umbilic Points at Infinity of Certain Algebraic Surfaces + abstract: | + In this lecture, we study the global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function \(f\) on the plane. We will prove that every umbilic point at infinity of the projective extension of these direction fields has a Poincar\'e-Hopf index equal to 1/2 and the topological type of a Lemon or a Monstar. As a consequence, we will provide a Poincar\'e-Hopf type formula for the graph of \(f\) pointing out that, if all umbilics are isolated, the sum of all indices of the principal directions at its umbilic points only depends upon the number of real linear factors of the homogeneous part of highest degree of \(f\). + This is a joint work with B. Guilfoyle. + start: + end: + speaker: Adriana Ortiz-Rodríguez (Universidad Nacional Autónoma de México, México) + - titulo: Perturbations of Hamiltonian foliations + abstract: | + At the International Congress of Mathematics held in Paris in 1900, D. Hilbert presented a list of 23 problems for the century. After more than one hundred years some of those problems are still unsolved. Among them is the 16th problem. This problem, in its second part, asks for the number and position of limit cycles (isolated periodic solutions) of planar polynomial differential equations. In the 70's, Y. Ilyashenko and V. I. Arnold proposed, in this context, to investigate what happens for systems infinitely close to Hamiltonians. This approach is known as the infinitesimal Hilbert 16th problem. More precisely, let \(F\in\mathbb{R}[x,y]\) a polynomial and \(\omega\) a polynomial 1-form. We consider the perturbation \(dF+\epsilon\omega=0\) of the Hamiltonian foliation \(dF=0\), where \(\epsilon\) is a small parameter. Let \(\gamma(z)\subset F^{-1}(z)\) a continuous family of regular periodic orbits of \(dF=0\). The displacement map with respect to this family \(\gamma(z)\) and with respect to the foliation \(dF+\epsilon\omega=0\) is an analytic function \(\Delta(z,\epsilon)=\epsilon^\mu M_\mu(z)+\cdots\). The first function \(M_\mu\) which does not vanish identically keeps information about limit cycles born from this perturbation. By passing to the complexification of \(F\), we can consider the orbit under monodromy of \(\gamma(z)\), which is a normal subgroup \(\mathcal{O}\) of the first homotopy group of the regular fiber \(F^{-1}(z)\). To understand the orbit under monodromy one must focus on the critical values of the function \(F\). To each critical value corresponds a vanishing cycle; the monodromy related to such cycle detects the traces of change in the topology due to the presence of the singular fiber. In non generic cases, not every cycle can be reached by the orbit under monodromy of the family of regular periodic orbits \(\gamma(z)\). Thus, to detect the cycles that are not reached one considers the quotient \(\frac{\mathcal{O}}{[\mathcal{O},\pi_1(F^{-1}(z),p_0)]}\) and defines a constant \(\kappa\), called orbit depth. It turns out that the function \(M_\mu\) is an iterated integral of length at most \(\kappa\). We stress that \(\kappa\) by its definition depends only on the topology of the regular fiber of \(F\), and on the orbit \(\mathcal{O}\) of \(\gamma(z)\), while the function \(M_\mu\) depends also on the perturbation given by \(\omega\). We will talk about this result and discuss some conjectures around this bound for non-generic polynomials. + start: + end: + speaker: Jessie Pontigo Herrera (Universidad Nacional Autónoma de México, México) + - titulo: Solutions of algebraic linear ordinary differential equations + abstract: | + A classical result of F. Klein states that, given a finite primitive group \(G\subseteq SL_2(\mathbb{C})\), there exists a hypergeometric equation such that any second order LODE whose differential Galois group is isomorphic to \(G\) is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this paper, we generalize this result. We show that, given a finite primitive group \(G\subseteq SL_n(\mathbb{C})\), there exist a positive integer \(d=d(G)\) and a standard equation such that any LODE whose differential Galois group is isomorphic to \(G\) is gauge equivalent, over a field extension \(F\) of degree \(d\), to an equation projectively equivalent to the pullback by a map in \(F\) of this standard equation. For \(n=3\), these standard equations can be chosen to be hypergeometric. + start: + end: + speaker: Camilo Sanabria (Universidad de Los Andes, Colombia) - sesion: Quantum symmetries organizadores: - nombre: César Galindo @@ -883,6 +1029,62 @@ mail: jplavnik@iu.edu - nombre: Leandro Vendramin mail: lvendramin@dm.uba.ar + charlas: + - titulo: 'On finite GK-dimensional Nichols algebras of diagonal type: rank 3 and Cartan type' + abstract: | + It was conjectured by Andruskiewitsch, Angiono and Heckenberger that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system; they did in fact prove this for braidings of affine type or when the rank is two. We shall review some tools developed with the intention of proving this conjecture positively, in a work of Angiono and the author, and exhibit the proof for the rank 3 case, as well as for braidings of Cartan type. + start: + end: + speaker: Agustín García Iglesias (Universidad Nacional de Córdoba, Argentina) + - titulo: A diagrammatic Carlsson-Mellit algebra + abstract: | + The \(A_{q,t}\) algebra was introduced by Carlsson and Mellit in their proof of the celebrated shuffle theorem, which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This algebra arises as an extension of two copies of the affine Hecke algebra by certain raising and lowering operators. Carlsson and Mellit constructed an action via plethystic operators on the space of symmetric functions which was then realized geometrically on parabolic flag Hilbert schemes by them and Gorsky. The original algebraic construction was then extended to an infinite family of actions by Mellit and shown to contain the generators of elliptic Hall algebra. However, despite the various formulations of \(A_{q,t}\), performing computations within it is complicated and non-intuitive. + In this talk I will discuss joint work with Matt Hogancamp where we construct a new topological formulation of \(A_{q,t}\) (at t=-1) and its representation as certain braid diagrams on an annulus. In this setting many of the complicated algebraic relations of \(A_{q,t}\) and applications to symmetric functions are trivial consequences of the skein relation imposed on the pictures. In particular, many difficult computations become simple diagrammatic manipulations in this new framework. This purely diagrammatic formulation allows us to lift the operators as certain functors, thus providing a categorification of the \(A_{q,t}\) action on the derived trace of the Soergel category. + start: + end: + speaker: Nicolle Gonzalez (The University of California Los Angeles, Estados Unidos) + - titulo: Quantum Symmetries in Conformal Field Theory + abstract: | + Quantum groups and Nichols algebras appear as symmetries in conformal field theories, as so-called non-local screening operators. I will start by giving a motivation of conformal field theory and explain how this leads from the analysis side to a modular tensor category, quite often one that is well known from algebra. I then explain my recent work that non-local screening operators generate Nichols algebras in this category, and also some recent work with T. Creutzig and M. Rupert that conclude a braided category equivalence between a certain physical model and the representation category of the smallest quantum group \(u_q(\mathfrak{sl}_2),\;q^4=1\). + start: + end: + speaker: Simon Lentner (Universität Hamburg, Alemania) + - titulo: Slack Hopf monads + abstract: | + Hopf monads are generalizations of Hopf algebras that bring to bear the rich theory of monads from category theory. Even though this generalization overarches many Hopf-like structures, quasi-Hopf algebras seem to be out of its reach. This talk reports on advances in the study of slack Hopf monads, including how these generalize quasi-Hopf algebras. (Joint work with A. Bruguieres and M. Haim). + start: + end: + speaker: Ignacio Lopez Franco (Universidad de la República, Uruguay) + - titulo: Interpolations of monoidal categories by invariant theory + abstract: | + In this talk, I will present a recent construction that enables one to interpolate symmetric monoidal categories by interpolating algebraic structures and their automorphism groups. I will explain how one can recover the constructions of Deligne for categories such as Rep(S_t), Rep(O_t) and Rep(Sp_t), the constructions of Knop for wreath products with S_t and GL_t(O_r), where O_r is a finite quotient of a discrete valuation ring, and also the TQFT categories recently constructed from a rational function by Khovanov, Ostrik, and Kononov. + start: + end: + speaker: Ehud Meir (University of Aberdeen, Escocia) + - titulo: Examples of module categories, the non-semisimple case + abstract: | + In this talk we present classical examples of exact indecomposable module categories over a finite non-semisimple rigid tensor category, as well as recently constructed new examples. + start: + end: + speaker: Adriana Mejía Castaño (Universidad del Norte, Colombia) + - titulo: Algebras in group-theoretical fusion categories + abstract: | + The categorical version of a module over a ring is the notion of a module category over a fusion category. We will first discuss what it means for a module category to be represented by an algebra and introduce the notion of Morita equivalence of algebras in fusion categories. Sonia Natale and Victor Ostrik described algebras representing the Morita equivalence classes in pointed fusion categories. We will explain how this result can be generalized to group-theoretical fusion categories, based on joint work with Monique Müller, Julia Plavnik, Ana Ros Camacho, Angela Tabiri, and Chelsea Walton. + start: + end: + speaker: Yiby Morales (Universidad de los Andes, Colombia) + - titulo: Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge + abstract: | + I will discuss recent work with T. Gannon in which we provide a ribbon tensor equivalence between the representation category of small quantum SL(2) at q=exp(pi*i/p) and the representation category of the triplet VOA at a corresponding central charge 1-6(p-1)^2/p. Such an equivalence was conjectured in work of Gainutdinov, Semikhatov, Tipunin, and Feigin from 2006. + start: + end: + speaker: Cris Negron (University of North Carolina, Estados Unidos) + - titulo: Frobenius-Schur indicators for some families of quadratic fusion categories + abstract: | + Quadratic categories are fusion categories with a unique non-trivial orbit from the tensor product action of the group of invertible objects. Familiar examples are the near-groups (with one non-invertible object) and the Haagerup-Izumi cate- gories (with one non-invertible object for each invertible object). Frobenius-Schur indicators are an important invariant of fusion categories generalized from the theory of finite group representations. These indicators may be computed for objects in a fusion category C using the modular data of the Drinfel’d center Z(C) of the fusion category, which is itself a modular tensor category. Recently, Izumi and Grossman provided new (conjectured infinite) families of modular data that include the modular data of Drinfel’d centers for the known quadratic fusion categories. We use this information to compute the FS indicators; moreover, we consider the relationship between the FS indicators of objects in a fusion category C and FS indicators of objects in that category’s Drinfel’d center Z(C). + start: + end: + speaker: Henry Tucker (University of California at Riverside, Estados Unidos) - sesion: Sistemas Dinámicos y Teoría Ergódica organizadores: - nombre: Jairo Bochi @@ -891,24 +1093,244 @@ mail: gelfert@im.ufrj.br - nombre: Rafael Potrie mail: rpotrie@cmat.edu.uy + charlas: + - titulo: Multiplicative actions and applications + abstract: | + In this talk, I will discuss recurrence problems for actions of the multiplicative semigroup of integers. Answers to these problems have consequences in number theory and combinatorics, such as understanding whether Pythagorean trios are partition regular. I will present in general terms the questions, strategies from dynamics to address them and mention some recent results we obtained. This is joint work with Anh Le, Joel Moreira, and Wenbo Sun. + start: + end: + speaker: Sebastián Donoso (Universidad de Chile, Chile) + - titulo: Actions of abelian-by-cyclic groups on surfaces + abstract: | + I'll discuss some results and open questions about global rigidity of actions of certain solvable groups (abelian by cyclic) on two dimensional manifolds. Joint work with Jinxin Xue. + start: + end: + speaker: Sebastián Hurtado-Salazar (University of Chicago, Estados Unidos) + - titulo: Polynomial decay of correlations of geodesic flows on some nonpositively curved surfaces + abstract: | + We consider a class of nonpositively curved surfaces and show that their geodesic flows have polynomial decay of correlations. This is a joint work with Carlos Matheus and Ian Melbourne. + start: + end: + speaker: Yuri Lima (Universidade Federal do Ceará, Brasil) + - titulo: Continuity of center Lyapunov exponents. + abstract: | + The continuity of Lyapunov exponents has been extensively studied in the context of linear cocycles. However, there are few theorems that provide information for the case of diffeomorphisms. In this talk, we will review some of the known results and explain the main difficulties that appear when trying to adapt the usual techniques to the study of center Lyapunov exponents of partially hyperbolic diffeomorphisms. + start: + end: + speaker: Karina Marín (Universidade Federal de Minas Gerais, Brasil) + - titulo: On tilings, amenable equivalence relations and foliated spaces + abstract: | + I will describe a family of foliated spaces constructed from tllings on Lie groups. They provide a negative answer to the following question by G.Hector: are leaves of a compact foliated space always quasi-isometric to Cayley graphs? Their construction was motivated by a profound conjecture of Giordano, Putnam and Skau on the classification, up to orbit equivalence, of actions of countable amenable groups on the Cantor set. I will briefly explain how these examples relate to the GPS conjecture. This is joint work with Fernando Alcalde Cuesta and Álvaro Lozano Rojo. + start: + end: + speaker: Matilde Martínez (Universidad de la República, Uruguay) + - titulo: Lyapunov exponents of hyperbolic and partially hyperbolic diffeomorphisms + abstract: | + If \(f\) is a diffeomorphism on a compact \(d\)-dimensional manifold \(M\) preserving the Lebesgue measure \(\mu\), then Oseledets Theorem tells us that almost every point has \(d\) Lyapunov exponents (possibly repeated): + $$\lambda_1(f,x)\leq\lambda_2(f,x)\leq\dots\leq\lambda_d(f,x).$$ + If furthermore $\mu$ is ergodic, then the Lyapunov exponents are independent of the point \(x\) (a.e.). We are interested in understanding the map + $$f\in Diff_{\mu}^r(M)\ \mapsto\ (\lambda_1(f),\lambda_2(f),\dots,\lambda_d(f)),\ r\geq 1.$$ + In general this map may be very complicated. However, if we restrict our attention to the set of Anosov or partially hyperbolic diffeomorphisms, then we can understand this map better. I will present various results related to the regularity, rigidity and flexibility of the Lyapunov exponents in this setting. + Some of the results presented are joint with C. Vasquez, F. Valenzuela, J. Yang and P. Carrasco. + start: + end: + speaker: Radu Saghin (Pontificia Universidad Católica de Valparaiso, Chile) + - titulo: Zero Entropy area preserving homeomorphisms on surfaces + abstract: | + We review some recent results describing the behaviour of homeomorphisms of surfaces with zero topological entropy. Using mostly techniques from Brouwer theory, we show that the dynamics of such maps in the sphere is very restricted and in many ways similar to that of an integrable flow. We also show that many of these restrictions are still valid for $2$-torus homeomorphisms. + start: + end: + speaker: Fabio Tal (Universidade de São Paulo, Brasil) + - titulo: Conjugacy classes of big mapping class groups + abstract: | + A surface \(S\) is big if its fundamental group is not finitely generated. To each big surface one can associate its mapping class group, \(\mathrm{Map}(S)\), which is \(\mathrm{Homeo}(S)\) mod isotopy. This is a Polish group for the compact-open topology. In this talk we study the action of \(\mathrm{Map}(S)\) on itself by conjugacy and characterize when this action has a dense or co-meager orbit. This is a joint work with Jesus Hernández Hernández, Michael Hrusak, Israel Morales, Anja Randecker and Manuel Sedano (arxiv.org/abs/2105.11282v2). + start: + end: + speaker: Ferrán Valdez (Universidad Nacional Autónoma de México, México) - sesion: Funciones especiales, polinomios ortogonales y teoría de aproximación organizadores: - nombre: Manuel Domínguez de la Iglesia mail: mdi29@im.unam.mx - nombre: Pablo Manuel Román mail: roman@famaf.unc.edu.ar + charlas: + - titulo: Asymptotic analysis of matrix orthogonal polynomials + abstract: | + In this talk we consider matrix orthogonal polynomials (MVOPs) on a finite interval of the real line, with Jacobi-type weights. We are particularly interested in the asymptotic behavior as the degree of the polynomials tends to infinity, that we study with the method of steepest descent applied to the corresponding Riemann-Hilbert problem, as described by Grünbaum, de la Iglesia and Martínez-Finkelshtein. We include examples motivated by group theory. + This is joint work with Arno Kuijlaars (KU Leuven, Belgium) and Pablo Román (Universidad de Córdoba, Argentina). + start: + end: + speaker: Alfredo Deaño (Universidad Carlos III de Madrid, España) + - titulo: Sobolev spaces on graded groups + abstract: | + Fractional Calculus is one of the areas where Special Functions naturally arise. We will characterize fractional Sobolev spaces of potential type on graded groups. The approach is based on the Littlewood-Paley g-function. This is a joint work with Pablo de Nápoli. + start: + end: + speaker: Rocío Diaz Martín (Universidad Nacional de Córdoba, Argentina) + - titulo: Elliptic Kac-Sylvester matrix from difference Lamé equation + abstract: | + Through a finite-dimensional reduction of the difference Lamé equation, an elliptic analog of the Kac-Sylvester tridiagonal matrix is found. We solve the corresponding finite discrete Lamé equation by constructing an orthogonal basis of eigenvectors for this novel elliptic Kac-Sylvester matrix. (Based on work in collaboration with Tamás Görbe.) + start: + end: + speaker: Jan Felipe van Diejen (Universidad de Talca, Chile) + - titulo: Sobolev Type orthogonal polynomials on the cone of revolution + abstract: | + We present the Sobolev Type orthogonal polynomials of several variables defined on the cone and on the surface of the cone as a particular case of the polynomials on a quadratic surface of revolution. Using the results from [1], [2] and [3], we show that this kind of polynomials satisfy some properties. + [1] L. Fernández, T. E Pérez, M. Piñar and Y. Xu. Krall-type orthogonal polynomials in several variables. Comp. Appl. Math. Vol 81. Pags 1519-1524. (2001). + [2] Y. Xu, Sobolev orthogonal polynomials defined via gradient on the unit ball, J. Approx. Theory. 152 (2008), 52--65. (2006). + [3] Y. Xu, Fourier series in orthogonal polynomials on a cone of revolution, arXiv:1905.07587 2019 + start: + end: + speaker: Herbert Dueñas Ruiz (Universidad Nacional de Colombia, Colombia) + - titulo: 'On spectral transformations of matrix orthogonal polynomials: some recent results' + abstract: | + In this contribution, we present some algebraic and analytic properties related to spectral transformations of orthogonality matrix measures on the unit circle, that constitute generalizations from well-known results on the scalar case. We focus our attention on the so-called Christoffel, Geronimus and Uvarov transformations, and deal with connection formulas, factorizations of block Hessenberg and CVM matrices, and relative asymptotics for the associated orthogonal matrix polynomials. This is a joint work with Edinson Fuentes. + start: + end: + speaker: Luis E. Garza Gaona (Universidad de Colima, México) + - titulo: Toda lattice, special functions and their matrix analogues + abstract: | + The classical Toda lattice is a model for a one-dimensional crystal. After a transformation in Flaschka coordinates there exists a Lax pair, for which the operator acts as a three-term recurrence operator. This gives a link to orthogonal polynomials, special functions and Lie algebra representations. In the case of orthogonal polynomials, the time dependence in the Toda lattice corresponds to deformation of the orthogonality measure by an exponential. The nonabelian Toda lattice is a generalisation of the Toda lattice for which matrix valued orthogonal polynomials play a similar role. We discuss matrix polynomials, and we discuss an explicit example of such a nonabelian Toda lattice. + start: + end: + speaker: Erik Koelink (Radboud Universiteit, Países Bajos) + - titulo: Multiple orthogonal polynomials with respect to hypergeometric functions + abstract: | + In this talk I will discuss on two new sets of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Confluent and Gauss' hypergeometric functions. The former on unbounded support and the latter with bounded support on the real line. It was recently shown in [3] that the latter are indeed random walk polynomials. In both cases, this type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total-positivity problems in enumerative combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indices lie on the step-line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. + References: + [1] H. Lima and A. Loureiro, Multiple orthogonal polynomials with respect to Gauss’ hypergeometric function, to appear in Stud. Appl. Math. arXiv:2001.06820. + [2] H. Lima and A. Loureiro, Multiple orthogonal polynomials associated with confluent hypergeometric functions, J. Approx. Theory, 260 (2020), pp 105484. + [3] A. Branquinho, J. E. Fernández-Díaz, A. Foulquié-Moreno and M. Mañas, Hypergeometric Multiple Orthogonal Polynomials and Random Walks, (2021) arXiv:2107.00770v2 + start: + end: + speaker: Ana Loureiro (University of Kent, Reino Unido) + - titulo: 'Poncelet-Darboux, Kippenhahn, and Szegö: projective geometry, matrices and orthogonal polynomials' + abstract: | + We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as Szegö polynomials. These results have connections to classical results from algebraic and projective geometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numerical ranges of a class of matrices; and Blaschke products and disk functions. + This is a joint work with Markus Hunziker, Taylor Poe, and Brian Simanek, all at Baylor University. + start: + end: + speaker: Andrei Martínez Finkelshtein (Baylor University, Estados Unidos, y Universidad de Almería, España) - sesion: Teoría algebraica de formas cuadráticas sobre anillos organizadores: - nombre: Hugo L. Mariano mail: hugomar@ime.usp.br - nombre: Alejandro Petrovich mail: apetrov@dm.uba.ar + charlas: + - titulo: Rings of Formal Power Series and Symmetric Real Semigroups + abstract: | + The aim of this talk is to present, firstly, a number of results on the relationship between the ring \(A = F[[G]]\) of formal power series with coefficients in a formally real (i.e., orderable) field \(F\) and exponents in the positive cone of a totally ordered abelian group \(G\), and the real semigroup \(G_{\!A}\) associated to the ring \(A\). One of the main results shows that the real semigroup \(G_{\!A}\) is a fan in the category of real semigroups if and only if the preorder \(\Sigma F^{2}\) of \(F\) is a fan in the sense of fields. On the other hand, in the general case (i.e., for an arbitrary formally real field of coefficients), the real semigroup \(G_{\!A}\) satisfies certain fundamental conditions formulated in terms of the specialization partial order \(\leadsto\) defined in the real spectrum \({\mathrm{Sper}}(A)\) of the ring \(A\). These properties led us to introduce a new class of real semigroups which we baptized symmetric real semigroups. + Next, we investigate the theory of symmetric real semigroups and prove several results on their structure, leading to the following: + I) Every finite symmetric real semigroup is realizable by a ring of formal power series. + II) There exists an infinite symmetric real semigroup (in fact, a fan) which is not realizable by any ring of formal power series. + start: + end: + 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. + start: + end: + speaker: Francisco Miraglia (Universidade de São Paulo, Brasil), joint with Hugo R. O. Ribeiro (USP) + - titulo: Ranges of functors and geometric classes + abstract: | + The representation problem for special groups, asking whether every reduced special group is isomorphic to the special group of a field, is an outstanding open problem in the realm of quadratic forms. The same is true for the corresponding problem about real semigroups, as well as Efrat's question which \(\kappa\)-structures arise as the Milnor \(K\)-theory of a field, are outstanding open problems in the realm of quadratic forms. All of these problems allow variants where one replaces isomorphism by elementary equivalence, e.g. by the question whether every reduced special group is elementarily equivalent to one coming from a field. + Motivated by these problems, we study the general question when the essential image of a functor can be axiomatized by \(\kappa\)-geometric sequents -a certain fragment of formulas of infinitary first order logic. + We observe that one can study this question via topos theory: Under mild hypotheses, functors between accessible categories (such as categories of models of first order theories) can be assumed to be induced by a \(\kappa\)-geometric morphism between classifying \(\kappa\)-toposes, notions first stdied by Espíndola. We show how this morphism can be factorized into a surjection, followed by a dense inclusion, followed by a closed inclusion, and explain what that means in terms of the involved theories. We arrive at very concrete axiomatizability criteria using this factorization. + All results and involved notions will be explained and backed up with examples. + start: + end: + speaker: Peter Arndt (Universität Düsseldorf, Alemania) + - titulo: Von Neumann Hull for Real Semigroups + abstract: | + The theory of Real Semigroups (RS) was created by M. Dickmann and A. Petrovich in the 2000's as a first order theory of real spectra of rings. It extends the concept of Reduced Special Group by using representation (or transversal representation) of dimension 2 form as primitive concept. + In this talk we will build the von Neumann Hull of a RS and describe its mains algebraic and categorical properties. As an application, we will prove a version of Marshall conjecture for real semigroups associated with (semi-real) rings. + start: + end: + speaker: Hugo Rafael de Oliveira Ribeiro (Universidade de São Paulo, Brasil), joint with Hugo L. Mariano (USP) + - titulo: K-theories, Graded Rings and Quadratic Forms + abstract: | + The uses of K-theoretic (and Boolean) methods in abstract theories of quadratic forms has been proved a very successful method, see for instance, these two papers of M. Dickmann and F. Miraglia: (1998) where they give an affirmative answer to Marshall's Conjecture, and (2003), where they give an affirmative answer to Lam's Conjecture. + These two central papers makes us take a deeper look at the theory of Special Groups by itself. This is not mere exercise in abstraction: from Marshall's and Lam's Conjecture many questions arise in the abstract and concrete context of quadratic forms. + There are some generalizations of Milnor's K-theory. In the quadratic forms context, the most significant one is the Dickmann-Miraglia's K-theory of Special Groups. It is a main tool in the proof of Marshall's and Lam's Conjecture. + In Marshall's paper (2006), he propose a new abstract theory of quadratic forms based on what he called a ``real reduced multiring'' and ``real reduced hyperfield''. This new theory has the advantage of brings new analogies with commutative algebra, and we developed and expand the details in Roberto, Ribeiro, Mariano (2020). It is even possible rewrite the axioms of special groups in a sort of "geometric manner" via hyperfields (Roberto, Ribeiro, Mariano (2021)). + Now, we will give another step in the ``marriage of multi structures and quadratic forms'' developing an appropriate K-theory for hyperfields. This new category generalizes simutaneously both Milnor's reduced and non-reduced K-theories and Dickmann-Miraglia's K-theory for special groups. + With these three K-theories on hands, it is desirable (or, at least, suggestive) the rise of an abstract enviroment that encapsule all them, and of course, provide an axiomatic approach to guide new extensions of the concept of K-theory in the context of the algebraic and abstract theories of quadratic forms. The inductive graded rings, introduced by M. Dickmann and F. Miraglia (2000) fits this purpose, and we finish this work showing that the K-theory of pre-special hyperfields is some kind of free inductive graded ring. + start: + end: + 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 - sesion: Lógica matemática organizadores: - nombre: Manuela Busaniche mail: mbusaniche@santafe-conicet.gov.ar - nombre: Marcelo Esteban Coniglio - mail: coniglio@cle.unicamp.br + mail: coniglio@cle.unicamp.br + charlas: + - titulo: Filter pairs and natural extensions of logics + abstract: | + We adjust the notion of finitary filter pair ( Finitary Filter Pairs and Propositional Logics, South American Journal of Logic 4(2)), which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality \(\kappa\), {where \(\kappa\) is a regular cardinal}. The corresponding new notion is called \(\kappa\)-filter pair. We show that any \(\kappa\)-filter pair gives rise to a logic of cardinality \(\kappa\) and that every logic of cardinality \(\kappa\) comes from a \(\kappa\)-filter pair. We use filter pairs to construct natural extensions for a given logic and work out the relationships between this construction and several others proposed in the literature ( A note on natural extensions in abstract algebraic logic, Studia Logica, 103(4); Constructing natural extensions of propositional logics, Studia Logica 104(6)). Conversely, we describe the class of filter pairs giving rise to a fixed logic in terms of the natural extensions of that logic. + start: + end: + speaker: Hugo Luiz Mariano (Universidade de São Paulo, Brasil), joint with P. Arndt (University of Düsseldorf, Germany) and D. C. Pinto (Federal University of Bahia, Brazil) + - titulo: A topological duality for monotone expansions of semilattices + abstract: | + The aim of this work is to present a topological duality for the variety of monotone semilattices. To do this, combining the duality developed by Celani and González for semilattices together with the approach given by Celani and the author for the study of monotone distributive semilattices, we introduce a category of multirelational spaces called \(m\)S-spaces and we prove that this is dually equivalent to the category of monotone semilattices with homomorphisms. Moreover, as an application of this duality we provide a characterization of congruences of (monotone) semilattices by means of lower Vietoris type topologies. It is worth mentioning that a key tool for developing this duality is the topological description of canonical extension of semilattices that we provide. This is a joint work with Ismael Calomino (CIC and Universidad Nacional del Centro) and William J. Zuluaga Botero (Universidad Nacional del Centro). + start: + end: + speaker: María Paula Menchón (Universidad Nacional del Centro de la Provincia de Buenos Aires, Argentina) + - titulo: Embeddability between orderings and GCH + abstract: | + We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete cardinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure hypothesis for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings. + start: + end: + speaker: Rodrigo de Alvarenga Freire (Universidade de Brasília, Brasil) + - titulo: Hilbert's tenth problem in rings of meromorphic functions + abstract: | + We will give a short survey on definability of the integers in rings of meromorphic functions, together with some ideas on the ingredients in the proofs and the reasons why the case of complex entire functions, as a ring, remain an open problem. We will then present two new evidences for the latter to be undecidable, one obtained with T. Pheidas, and the other one with D. Chompitaki, N. García-Fritz, H. Pasten and T. Pheidas. + start: + end: + speaker: Xavier Vidaux (Universidad de Concepción, Chile) + - titulo: Existentially closed measure preserving actions of the free group. + abstract: | + In joint work with Ward Henson and Tomás Ibarlucía, we prove that the theory of atomless probability algebras expanded with a measure preserving action of the free group \(F_k\) has a model companion for any \(k \geq 1\) and the corresponding theory is stable. We build explicitly examples of existentially closed models. Our approach also applies to a reasonably large class of \(\aleph_0\)-categorical superstable theories. + start: + end: + speaker: Alexander Berenstein (Universidad de los Andes, Colombia) + - titulo: Completitud estándar fuerte (fuerte finita) para lógicas de S5-modales de Łukasiewicz + abstract: | + En esta charla mostraremos una versión algebraica de teoremas de completitud estándar fuerte finita para lógicas de S5-modales de Łukasiewicz y algunas de sus extensiones axiomáticas de interés. Para esto mostraremos que la variedad de la MV-álgebras monádicas están generadas como cuasivariedad por cierta álgebra funcional estándar. Lo mismo se hará para ciertas extensiones axiomáticas de interés. Por último agregaremos una regla infinitaria y mostraremos la completitud estándar fuerte para esa lógica usando nuevamente técnicas algebraicas. + start: + end: + speaker: José Patricio Díaz Varela (Universidad Nacional del Sur, Argentina) + - titulo: Distributivity, Modularity, and Natural Deduction + abstract: | + On the one hand, from a logical point of view, we will pay attention to discussions around the notion of distributivity in the nineteenth century world of the algebra of logic. In particular, we will consider some passages from the writings of Ernst Schršoder and Charles Peirce. On the other hand, from an algebraic point of view, we will trace the origins of the notion of modularity in lattice theory. This we will do examining some writings of Richard Dedekind. Moreover, we will include some remarks on the different ways to consider the notions of modularity and distributivity in the context of a join semilattice. In the end, paying special attention to the disjunction elimination rule in natural deduction, we will include considerations on those notions in the context of the conjunction-disjunction-negation fragment of propositional intuitionistic logic. + start: + end: + speaker: Rodolfo C. Ertola-Biraben (Universidad de Campinas, Brasil) + - titulo: Interpolation and Beth definability for conic idempotent Full Lambek calculus + abstract: | + The Full Lambek calculus forms the basic system for substructural logics, examples of which include relevance, linear, many-valued, intuitionistic and classical logic. We focus on substructural logics that satisfy contraction, mingle and the rule of conicity. + The algebraic semantics of substructural logics are residuated lattices and they have an independent history in the context of order algebra; they were first introduced by Ward and Dilworth as tools in the study of ideal lattices of rings. Residuated lattices have a monoid and a lattice reduct, as well as division-like operations; examples include Boolean algebras, lattice-ordered groups and relation algebras. They are also connected to mathematical linguistics and computer science (for example pointer management and memory allocation). We focus on idempotent conic residuated lattices, which are related to algebraic models of relevance logic, and which extend the class of idempotent linear ones. After establishing a decomposition result for this class, we show that it has the strong amalgamation property, and extend the result to the variety generated by this class; this implies that the corresponding logic has the interpolation property and the Beth definability property. + start: + end: + speaker: Nick Galatos (University of Denver, Estados Unidos), joint with Wesley Fussner + - titulo: Lindström theorems in graded model theory + abstract: | + Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this talk I will present some first steps towards an abstract formulation of this model theory. I will give a general notion of abstract logic based on many-valued models and discuss six Lindstr\"{o}m-style characterizations of maximality of first-order logics in terms of metalogical properties such as compactness, abstract completeness, the L\"{o}wenheim-Skolem property, the Tarski union property, and the Robinson property, among others. The results have been obtained in a joint work with Guillermo Badia. + start: + end: + speaker: Carles Noguera i Clofent (Czech Academy of Sciences, República Checa) + - titulo: TBA + abstract: | + TBA + start: + end: + speaker: Xavier Caicedo (Universidad de los Andes, Colombia) - sesion: Operator Algebras organizadores: - nombre: Fernando Abadie @@ -917,6 +1339,56 @@ mail: alcides.buss@ufsc.br - nombre: Damián Ferraro mail: damian@cmat.edu.uy + charlas: + - titulo: Leavitt path algebras and graph C*-algebras associated to separated graphs + abstract: | + A separated graph is a pair \((E,C)\), where \(E\) is a directed graph, \(C=\bigsqcup _{v\in E^ 0} C_v\), and \(C_v\) is a partition of \(r^{-1}(v)\) (into pairwise disjoint nonempty subsets) for every vertex \(v\). In recent years, separated graphs have been used to provide combinatorial models of several structures, often related to dynamical systems. This can be understood as a generalization of the common use of usual directed graphs in symbolic dynamics. I will survey some of these developments, including the failure of Tarski's dichotomy in the setting of topological actions, the construction of a family of ample groupoids with prescribed type semigroup, and the modeling of actions on the Cantor set. + start: + end: + speaker: Pere Ara (Universitat Autònoma de Barcelona, España) + - titulo: A non-commutative topological dimension + abstract: | + Due to the Gelfand duality, C*-algebras are regarded as non-commutative topological spaces. This approach has sparked fundamental ideas in the structure and classification theory of C*-algebras. In this talk, I will explain a notion of non-commutative covering dimension and its impact in the classification programme of C*-algebras. + start: + end: + speaker: Jorge Castillejos (Instituto de Ciencias Matemáticas ICMAT, España) + - titulo: Isotropy algebras and Fourier analysis for regular inclusions + abstract: | + Given a regular inclusion \(A\subseteq B\) of C*-algebras, with \(A\) abelian, we will define the notion of ``isotropy algebra'' for any given point \(x\) in the spectrum of \(A.\) We will then compare this notion with the usual notion of isotropy groups in the context of groupoid C*-algebras, showing that the isotropy algebra relative to the full groupoid C*-algebra is naturally isomorphic to the full group C*-algebra of the corresponding isotropy group. Based on the slightly generalized notion of ``isotropy module'', we will then introduce a method to analyze the structure of \(B\) paralleling the classical Fourier analysis. This talk is based on the paper [Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids, arXiv:1901.09683 [math.OA] [v3] Tue, 1 Jun 2021], joint with David Pitts. + start: + end: + speaker: Ruy Exel (Universidade Federal de Santa Catarina, Brasil) + - titulo: Toeplitz algebras of semigroups and their boundary quotients + abstract: | + In general there are many distinct C*-algebras generated by isometric representations of a left cancellative monoid \(P\). Two distinguished examples are the Toeplitz C*-algebra \(\mathcal{T}_\lambda(P)\) generated by the left regular representation of \(P\) on \(\ell^2(P)\), and its boundary quotient \(\partial\mathcal{T}_\lambda(P)\). I will report on recent work with M. Laca, in which we define a universal Toeplitz C*-algebra \(\mathcal{T}_u(P)\) for a submonoid of a group that is canonically isomorphic to Li's semigroup C*-algebra when independence holds and is the universal analogue of \(\mathcal{T}_\lambda(P)\) also when independence fails. I plan to focus mainly on boundary quotients, explaining why the covariance algebra of the canonical product system over \(P\) with one-dimensional fibres may be regarded as the universal analogue of \(\partial\mathcal{T}_\lambda(P)\). I will give a concrete presentation of the full boundary quotient using a new notion of foundation sets, and give sufficient conditions on \(P\) for \(\partial\mathcal{T}_\lambda(P)\) to be purely infinite simple. If time permits, I will also address faithfulness of representations at the level of Toeplitz algebras. + start: + end: + speaker: Camila Fabre (Victoria University of Wellington, Nueva Zelanda) + - titulo: Levantamientos de mapas completamente positivos + abstract: | + Un resultado clásico de Choi-Effros afirma que los mapas nucleares completamente positivos entre C*-álgebras siempre tienen levantamientos completamente positivos. Por otro lado, hay muchos ejemplos que muestran que los mapas no nucleares no siempre admiten levantamientos completamente positivos. En esta charla, que está basada en un trabajo conjunto con Marzieh Forough y Klaus Thomsen, probamos que aún si el mapa no es nuclear, siempre existe una familia continua de levantamientos que es asintóticamente completamente positiva. Cuando todas las C*-álgebras están munidas de acciones continuas de un grupo localmente compacto, y el mapa completamente positivo es equivariante, también probamos que la familia continua de levantamientos puede construirse de modo que además sea asintóticamente equivariante, uniformemente en subconjuntos compactos del grupo. Si todas las C*-álgebras y el mapa dado son unitales, los levantamientos no pueden en general construirse de forma que también sean unitales, a menos que el grupo sea promediable. De hecho, esta propiedad de los levantamientos caracteriza la amenabilidad del grupo actuante. + start: + end: + speaker: Eusebio Gardella (Universidad de Münster, Alemania) + - titulo: Continuous orbit equivalence and full groups of ultragraph C*-algebras + abstract: | + In this talk, we describe ultragraphs, their associated edge shift spaces (which generalize SFT for infinite alphabets), and their associated C*-algebras and groupoids. We present results regarding continuous orbit equivalence of Deaconu-Renault systems and full groups associated with groupoids. To finish, we describe how to apply these results for ultragraph algebras. + start: + end: + speaker: Daniel Gonçalves (Universidade Federal de Santa Catarina, Brasil) + - titulo: Revisiting quantum mechanics via quantales and groupoid C*-algebras + abstract: | + It is well known that operator algebras are the mathematical language of algebraic quantum mechanics and algebraic quantum field theory. In particular, von Neumann algebras cater for a natural formalization of Heisenberg's ``picture'' of quantum mechanics, which rests on interpreting physical observables as being time-dependent self-adjoint operators, to which von Neumann explicitly added, still in his twenties, that physical measurements should correspond to projection operators, thus producing the first mathematically thorough account of the foundations of quantum theory. However, from a conceptual point of view deep problems remained. These, in essence, have led to the seemingly unstoppable proliferation of interpretations and modifications of quantum mechanics, despite which, to this day, the ``measurement problem'' remains largely unsolved. In this talk I describe ongoing work whose aim is to address this by means of a definition of space of measurements whose open sets correspond to the finite pieces of classical information that can be extracted during a measurement. The interplay between C*-algebras, locally compact étale groupoids, and quantales plays an important role. The main purpose of the talk is to explain the main ideas surrounding measurement spaces along with related facts about groupoid C*-algebras and Fell bundles. + start: + end: + speaker: Pedro Resende (Instituto Superior Técnico, Portugal) + - titulo: Permanence properties of verbal products and verbal wreath products of groups + abstract: | + Given a family of groups, the direct sum and the free product provide useful ways of constructing new groups out of it. Verbal products of groups, which were defined in the fifties, give many new operations in groups which share some common features with the direct sum and the free product. Arguably, these products have not been studied much in recent years and they might have been overlooked in geometric and measurable group theory. + In this talk we will review these products, give some examples and show that several group theoretical properties that are of much interest in operator algebras like amenability, Haagerup property, property (T), exactness, soficity, Connes' embedability, etc. are preserved under some of these products. Afterwards, we will explain how to define verbal wreath products between two groups, (these are semi-direct products that resemble the restricted wreath product), and we will show that the Haagerup property, soficity, Connes' embeddability and other metric approximation properties are preserved under taking verbal wreath products. Finally we will give some applications of them. + start: + end: + speaker: Román Sasyk (Universidad de Buenos Aires, Argentina) - sesion: Teoría de Números organizadores: - nombre: María de los Ángeles Chara @@ -925,6 +1397,50 @@ mail: guillermoa.mantillas@konradlorenz.edu.co - nombre: Amalia Pizarro mail: amalia.pizarro@uv.cl + charlas: + - titulo: Serre's modular conjecture. + abstract: | + In this talk we will recall the formulation of Serre's modular conjecture, and we will explain how some strong new modularity results can be used to give a simplified proof of it. This is a joint work with Luis Dieulefait. + start: + end: + speaker: Ariel Pacetti (Universidade de Aveiro, Portugal) + - titulo: Monogenic and binary number fields of small degree + abstract: | + We show that a positive proportion of cubic and quartic number fields are not monogenic, and not just for local reasons. We also show that a positive proportion of quartic number fields are not binary, i.e. they don't arise as the invariant order attached to a binary quartic form. To prove these results, we study rational points (and the lack thereof) in a family of elliptic curves. Joint work with Levent Alpoge and Manjul Bhargava. + start: + end: + speaker: Ari Shnidman (Einstein Institute of Mathematics, Israel) + - titulo: The integral trace form and shape as complete invariants for real Sn number fields + abstract: | + In this talk we will discuss two invariants attached to a number field: the integral quadratic form associated to its trace paring and its shape. We prove that, as long as we restrict the ramification, these invariants are in fact strong enough to completely determine the isomorphism class of a field within the family of totally real Sn-number fields; as a byproduct of the method we are able to describe the automorphism group of the integral trace form for such fields. The chief ingredient in the proofs is a linear algebra gadget for quadratic forms we called “Casimir pairings”. They generalize the Casimir element from the theory of Lie algebras as well as the usual inner product of 1-forms in Riemannian geometry. For trace forms of number fields (and more generally of étale algebras), I will explain how the main usefulness of Casimir pairings lies in their functorial properties which make them compatible with both Galois-étale theory and base changes. This is joint work with Guillermo Mantilla-Soler. + start: + end: + speaker: Carlos A. Rivera (University of Washington, Estados Unidos) + - titulo: Upper bounds on counting number fields + abstract: | + I will present a general upper bound for the number of number fields of fixed degree and bounded discriminant. The method refines that of Ellenberg and Venkatesh, and improves upon their bound and that of Couveignes. This is joint work with Robert Lemke Oliver. + start: + end: + speaker: Frank Thorne (University of South Carolina, Estados Unidos) + - titulo: Sums of certain arithmetic functions over \(\mathbb{F}_q[T]\) and symplectic distributions + abstract: | + In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function \(d_k(f)\) over short intervals and over arithmetic progressions for the function field \(\mathbb{F}_q[T]\) to certain integrals over the ensemble of unitary matrices when \(q \rightarrow \infty\). We study two problems: the average over all the monic polynomials of fixed degree that yield a quadratic residue when viewed modulo a fixed monic irreducible polynomial \(P\), and the average over all the monic polynomials of fixed degree satisfying certain condition that is analogous to having an argument (in the sense of complex numbers) lying at certain specific sector of the unit circle. Both problems lead to integrals over the ensemble of symplectic matrices when \(q \rightarrow \infty\). We also consider analogous questions involving convolutions of the von Mangoldt function. This is joint work with Vivian Kuperberg. + start: + end: + speaker: Matilde Lalín (Université de Montréal, Canadá) + - titulo: Congruences satisfied by eta quotients + abstract: | + The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results given by Ahlgren and Ono for the partition function and by Treneer for more general Fourier coefficients state the existence of infinitely many families of congruences. We give an algorithm for computing explicit instances of such congruences for eta-quotients, and we illustrate our method with a few examples. + Joint work with Nathan Ryan, Zachary Scherr and Stephanie Treneer. + start: + end: + speaker: Nicolás Sirolli (Universidad de Buenos Aires, Argentina) + - titulo: p-adic asymptotic distribution of CM points + abstract: | + A CM point in the moduli space of complex elliptic curves is a point representing an elliptic curve with complex multiplication. A classical result of William Duke (1988), complemented by Laurent Clozel and Emmanuel Ullmo (2004), states that CM points become uniformly distributed on the moduli space when we let the discriminant of the underlying ring of endomorphisms of these elliptic curves go to infinity. Since CM points are algebraic, it is possible to study p-adic analogues of this phenomenon. In this talk I will present a description of the p-adic asymptotic distribution of CM points in the moduli space of p-adic elliptic curves. This is joint work with Ricardo Menares (PUC, Chile) and Juan Rivera-Letelier (U. of Rochester, USA). + start: + end: + speaker: Sebastián Herrero (Pontificia Universidad Católica de Valparaíso, Chile) - sesion: Dinámica de grupos organizadores: - nombre: Juan Alonso @@ -935,12 +1451,129 @@ mail: shurtados@uchicago.edu - nombre: Cristóbal Rivas mail: cristobal.rivas@usach.cl + charlas: + - titulo: Orbit equivalence and dynamical properties of minimal group actions on the Cantor set + abstract: | + Two topological dynamical systems \((X, T, G)\) and \((Y,S,\Gamma)\) are said to be (topologically) orbit equivalent if there exists a homeomorphism \(h:X\to Y\) sending \(T\)-orbits onto \(S\)-orbits. A general question in the framework of orbit equivalence is how the dynamical properties of a given system are related to its orbit equivalence class. In this talk we will discuss about what are the natural restrictions that arise from being orbit equivalent, in the case of minimal shift actions on the Cantor set given by countable amenable groups. We will also discuss about the problem of realization of Choquet simplices as sets of invariant measures of topological dynamical systems and see how this is connected to the problem of classification up to orbit equivalence. + start: + end: + speaker: Paulina Cecchi (Universidad de Chile, Chile) + - titulo: Random Walks and CAT(0) Cube Complexes + abstract: | + Let G be a group acting on a finite dimensional CAT(0) cube complex X. By studying equivariant maps from the Furstenberg-Poisson boundary to the Roller boundary, we deduce a variety of phenomena concerning the push-forward of the random walk from G to an orbit in X. Under mild and natural assumptions, we deduce positivity of the drift, sublinear tracking, and a central limit theorem. Along the way we prove that regular elements are plentiful and establish a homeomorphism between the boundary of the contact graph of X with a special subset of the Roller boundary called the regular points. This is joint work with Jean Lécureux and Frédéric Mathéus. + start: + end: + speaker: Talia Fernós (The University of North Carolina at Greensboro, Estados Unidos) + - titulo: Optimal regularity of mapping class group actions on the circle + abstract: | + We prove that for each finite index subgroup \(H\) of the mapping class group of a closed hyperbolic surface, and for each real number \(r>0\) there does not exist a faithful \(C^{1+r}\)--action of \(H\) on a circle. (Joint with Thomas Koberda and Cristobal Rivas) + start: + end: + speaker: Sang-hyun Kim (Korea Institute for Advanced Study, Corea del Sur) + - titulo: Hyperbolic groups acting on their boundaries + abstract: | + A hyperbolic group acts on its Gromov boundary by homeomorphisms. In recent joint work with Jason Manning, we show these actions are topologically stable whenever the boundary is a sphere: any small perturbation of the action is semi-conjugate to the original action. This is also true for free groups, with cantor set boundary, and in ongoing work we are investigating the general case. In my talk, I will explain some of the strategy of the proof and motivation for the problem. + start: + end: + speaker: Kathryn Mann (Cornell University, Estados Unidos) + - titulo: 'Locally moving groups acting on the real line I: C^1 actions' + abstract: | + Given a group G, we are interested in describing the possible actions of G on the real line, that is its representations into the group of homeomorphisms of the real line. In this series of two talks, we present a setup that allows to give a satisfactory picture for a class of groups, which arise as sufficiently rich subgroups of the group of homeomorphisms of the line, called locally moving groups. This class contains various finitely generated groups acting on the line: a famous example is Thompson’s group F. + In this first talk, we will focus on action by C^1-diffeomorphisms. In particular we will see that if G is a locally moving group of homeomorphisms on the line, then every faithful minimal action of G on the line is topologically conjugate to its natural defining action. + In the second talk, we will focus on actions that are only by homeomorphisms, that turn out to be much more flexible: for example we will explain that Thompson’s group F admits a continuum of exotic minimal faithful C^0 actions on the line. Nevertheless we will see that such actions have a very special structure: this will lead us to introduce the notion of R-focal action, a class of actions on the line that can be suitably encoded by certain actions on planar real trees. As an application we will obtain a local rigidity result: for a vast class of locally moving groups, all small perturbations of the natural defining actions are semi-conjugate to it. + The talks are based on joint work of the speakers with J. Brum and C. Rivas. + start: + end: + speaker: Nicolás Matte Bon (Université de Lyon, Francia) + - titulo: 'Locally moving groups acting on the real line II: exotic actions' + abstract: | + Given a group G, we are interested in describing the possible actions of G on the real line, that is its representations into the group of homeomorphisms of the real line. In this series of two talks, we present a setup that allows to give a satisfactory picture for a class of groups, which arise as sufficiently rich subgroups of the group of homeomorphisms of the line, called locally moving groups. This class contains various finitely generated groups acting on the line: a famous example is Thompson’s group F. + In this first talk, we will focus on action by C^1-diffeomorphisms. In particular we will see that if G is a locally moving group of homeomorphisms on the line, then every faithful minimal action of G on the line is topologically conjugate to its natural defining action. + In the second talk, we will focus on actions that are only by homeomorphisms, that turn out to be much more flexible: for example we will explain that Thompson’s group F admits a continuum of exotic minimal faithful C^0 actions on the line. Nevertheless we will see that such actions have a very special structure: this will lead us to introduce the notion of R-focal action, a class of actions on the line that can be suitably encoded by certain actions on planar real trees. As an application we will obtain a local rigidity result: for a vast class of locally moving groups, all small perturbations of the natural defining actions are semi-conjugate to it. + The talks are based on joint work of the speakers with J. Brum and C. Rivas. + start: + end: + speaker: Michele Triestino (Université de Bourgogne, Francia) + - titulo: Projective manifolds, hyperbolic manifolds and the Hessian of Hausdorff dimension + abstract: | + Let \(\Gamma\) be the fundamental group of a closed (real) hyperbolic \(n\)-manifold \(M.\) We study the second variation of the Hausdorff dimension of the limit set of convex co-compact morphisms acting on the complex-hyperbolic space \rho:\Gamma\to Isom(\mathbb H^n_\mathbb C), obtained by deforming a discrete and faithful representation of \(\Gamma\) that preserves a totally geodesic (and totally real) copy of the real-hyperbolic space \mathbb H^n_\mathbb R\subset\mathbb H^n_\mathbb C. This computation is based on the study of the space of convex projective structures on \(M\) and a natural metric on it induced by the Pressure form. This is joint work with M. Bridgeman, B. Pozzetti and A. Wienhard. + start: + end: + speaker: Andrés Sambarino (Sorbonne Université, Francia) - sesion: Functional Differential Equations and its Applications organizadores: - nombre: Pablo Amster mail: pamster@dm.uba.ar - nombre: Gonzalo Robledo mail: grobledo@uchile.cl + charlas: + - titulo: Coupled reaction-diffusion and difference system with nonlocal dispersal term and implicit time delay + abstract: | + We consider a class of biological models represented by a coupled system between reaction-diffusion and difference equations (renewal equation) with nonlocal dispersal term and implicit time delay. The difference equation generally arises, using the characteristic method, from an age-structured partial differential system. We start by studying the existence, uniqueness, positivity, and boundedness of solutions. We then investigate the stability analysis and obtain a threshold condition for the global asymptotic stability of the trivial steady state by using a Lyapunov functional. We also obtain a sufficient condition for the existence and uniqueness of a positive steady-state by using the method of lower and upper solutions. In the case of an unbounded domain, we study the existence of monotonic planar traveling wave fronts connecting the extinction state to the uniform positive state. The corresponding minimum wave speed is also obtained. In addition, we investigate the effect of the parameters on this minimum wave speed and we give a detailed analysis of its asymptotic behavior. + start: + end: + speaker: Mostafa Adimy (Institut de la Recherche en Informatique et Automatique, Francia) + - titulo: Periodic positive solutions of superlinear delay equations via topological degree + abstract: | + We present a joint work with Pablo Amster (Universidad de Buenos Aires, Argentina) and Julián Haddad (Universidade Federal de Minas Gerais, Brazil), which deals with an extension to delay problems of some recent results by G. Feltrin and F. Zanolin. They obtained existence and multiplicity results for positive periodic solutions to nonlinear differential equations of the form + \[u''(t)=f(t,u(t),u'(t))\] + with periodic or Neumann boundary conditions and special assumptions on $f$. We obtain analogous results for a delay problem of the type + \[u''(t)=f(t,u(t),u(t-\tau),u'(t))\] + Our approach, as well as that of Feltrin and Zanolin, is topological and based on the coincidence degree introduced by J. Mawhin. + start: + end: + speaker: Pierluigi Benevieri (Universidade de Sao Paulo, Brasil) + - titulo: A Krasnoselskii relatedness principle for delay differential equations + abstract: | + In this talk, we shall present an \textbf{Abstract} formulation of a duality principle established by Krasnoselskii. Under appropriate conditions, it shall be shown that, if the solutions of a nonlinear functional equation can be obtained by finding fixed points of certain operators in possibly different Banach spaces, then these operators have the same topological index. An application to delay differential equations shall be given. This is a joint work with P. Amster. + start: + end: + speaker: Julian Epstein (Universidad de Buenos Aires, Argentina) + - titulo: Stability and boundedness of solutions of retarded equations + abstract: | + Generalized ODEs were introduced by J. Kurzweil in 1957 and are known to encompass several other types of equations. In this talk, we explore converse Lyapunov theorems for this type of equations. In particular, we derive results for certain integral forms of measure functional differential equations. We also relate Lyapunov stability to the boundedness of solutions of these equations which have non--absolute integrable right--hand sides. + start: + end: + speaker: Márcia Federson (Universidade de Sao Paulo, Brasil) + - titulo: Global bifurcation-like Theorems in presence of non-vanishing Spectral Flow + abstract: | + Given a one-parameter family of functions \(f_t:H \to \mathbb R\) where \(f_t (0) = 0, \nabla f_t (0) = 0\) and \(H\) is a real Hilbert space, the Spectral Flow of the Hessian of \(f\) was related recently to local bifurcation results by Fitzpatrick, Pejsachowicz and Waterstraat. While this invariant is finer than the topological index, the existent results are of local nature, in contrast to the global bifurcation theorem of Krasnoselski and Rabinowitz. We prove a global bifurcation theorem for “target values” of \(f\) under Spectral Flow hypothesis. + start: + end: + speaker: Julian Haddad (Universidade Federal de Minas Gerais, Brasil) + - titulo: On the Stability and the Existence and Non-existence of \(T-\)Periodic Solutions for Nonlinear Delayed Differential Equations with Friction and \(\varphi\)-Laplacian + abstract: | + Let us consider the following problem + \[(\varphi(x'(t) ))' + h(x(t),x'(t))x'(t)+ g(x(t-r)) = p(t), \ \ \ t\in [0,\infty),\] + where \(\varphi:\mathbb{R} \rightarrow \mathbb{R}\) is an increasing homeomorphism such that \(\varphi(0)=0\), \(r\) is a positive constant, \(g:\mathbb{R} \rightarrow \mathbb{R}\) is a continuous differentiable function, and \(h:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}\) and \(p:[0,\infty) \rightarrow \mathbb{R}\) are continuous functions such that \(p\) is \(T\)-periodic in \(t\) for \(T\) a positive constant. + Using Lyapunov-Krasovskii functional and under appropriate assumptions we obtain new results on the global stability, boundedness of solutions, existence and non-existence of \(T-\)periodic solutions. This is a joint work with P. Amster and D.P. Santos. + start: + end: + speaker: Mariel Paula Kuna (Universidad de Buenos Aires, Argentina) + - titulo: Linearized instability for neutral FDEs + abstract: | + In this talk, we will give a brief overview of a class of equations called neutral functional differential equations with state-dependent delays, describing some important applications. After this, we will show some recents results in the area, and we will present a principle of linearized instability for these equations.This is a joint work with Professor Bernhard Lani-Wayda. + start: + end: + speaker: Jaqueline Godoy Mesquita (Universidade de Brasilia, Brasil) + - titulo: Existence of bifurcation point for impulsive differential equations via generalized ordinary differential equations + abstract: | + This is a joint work with Professors M. Federson and J. Mawhin. In this work, we establish conditions for the existence of a bifurcation point with respect to the trivial solution of a generalized ordinary differential equation, whose integral form displays the nonabsolute Kurzweil integral. The main tools employed here are the coincidence degree theory and an Arzela-Ascoli-type theorem for regulated functions. We also present applications to impulsive differential equations. + start: + end: + speaker: Maria Carolina Mesquita (Universidade de Sao Paulo, Brasil) + - titulo: A Lazer-Leach type result for functional-differential equations at resonance + abstract: | + In this talk we will discuss some recent results regarding Lazer-Leach type conditions for the existence of periodic solutions to systems of functional-differential equations at resonance. We consider even-dimensional kernels and general delays using Coincidence Degree Theory. + start: + end: + speaker: Arturo Sanjuán (Universidad Distrital Francisco José de Caldas, Colombia) + - titulo: On the nonlinearly determined wavefronts for the Mackey-Glass type diffusive equations + abstract: | + We study the Mackey-Glass type monostable delayed reaction-diffusion equation with a unimodal birth function \(g(u)\). This model, designed to describe evolution of single species populations, is considered in the presence of the weak Allee effect (\(g(u_0)>g'(0)u_0\) for some \(u_0>0\)). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. By considering the particular case of Nicholson's diffusive equation, we also discuss other situation leading to the appearance of nonlinearly determined wavefronts. This is a joint work with Karel Hasík, Jana Kopfová and Petra Nábělková, Mathematical Institute, Silesian University, Czech Republic. + start: + end: + speaker: Sergei Trofimchuk (Universidad de Talca, Chile) - sesion: Problemas Variacionales y Ecuaciones Diferenciales Parciales organizadores: - nombre: Judith Campos Cordero @@ -949,6 +1582,80 @@ mail: dhenao@mat.puc.cl - nombre: Dora Cecilia Salazar Lozano mail: dcsalazarl@unal.edu.co + charlas: + - titulo: Regularity for \(C^{1,\alpha}\) interface transmission problems + abstract: | + Transmission problems originally arose in elasticity theory, and they are nowadays of great interest due to its many applications in different areas in science. Typically, in these problems, there is a fixed interface where solutions may change abruptly, and the primary focus is to study their behavior across this surface. In this talk, we will discuss existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with \(C^{1,\alpha}\) interfaces. For this, we develop a novel geometric stability argument based on the mean value property. + These results are part of my PhD dissertation, and they are joint work with Luis A. Caffarelli and Pablo R. Stinga. + start: + end: + speaker: María Soria-Carro (University of Texas, Estados Unidos) + - titulo: A Non-variational Approach for the Porous Medium Equation + abstract: | + We extend the Krylov-Safonov theory and the corresponding H\"older estimates for the viscosity solutions of a type porous medium equation with non-variational structure. This work is in collaboration with Makson Santos (CIMAT-Guanajuato). + start: + end: + speaker: Héctor Andrés Chang Lara (Centro de Investigación en Matemáticas, México) + - titulo: Blow up for the Keller-Segel system in the critical mass case + abstract: | + We consider the Keller-Segel system in the plane with an initial condition with suitable decay and mass \(8 \pi\), which corresponds to the threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function \(u_0\) with mass \(8 \pi\) such that for any initial condition sufficiently close to \(u_0\) and the same mass, the solution is globally defined and blows-up in infinite time. We also find the profile and rate of blow-up. This result answers affirmatively the question of the nonradial stability raised by Ghoul and Masmoudi (2018). This is a joint work with M. del Pino, M. Musso and J. Wei. + start: + end: + speaker: Juan Dávila (University of Bath, Inglaterra y Universidad de Antioquia, Colombia) + - titulo: Critical polyharmonic systems and optimal partitions + abstract: | + In this talk, we establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in \(\mathbb{R}^N\) which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to \(-\infty\). We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of \(\mathbb{R}^N\). Moreover, we give a detailed description of the shape of these domains. + This is a joint work with Mónica Clapp and Alberto Saldaña. + start: + end: + speaker: Juan Carlos Fernández Morelos (Universidad Nacional Autónoma de México, México) + - titulo: Blowing up solutions for sinh-Poisson type equations on pierced domains + abstract: | + We consider sinh-Poisson type equations (either Liouville or mean field form) with variable intensities and Dirichlet boundary condition on a pierced domain \(\Omega_\varepsilon:=\Omega\setminus \cup_{i=1}^m B(\xi_i,\varepsilon_i)\), where \(\Omega\) is a smooth bounded domain in \(\text{I\!R}^2\) which contains the points \(\xi_i\), \(i=1,\dots,m\), \(\varepsilon_i=\varepsilon_i(\varepsilon)>0\), \(i=1,\dots,m\), depending on some parameter \(\varepsilon>0\) small enough. Given \(m\) different points \(\xi_i\), \(i=1,\dots,m\), we have found suitable radii \(\varepsilon_i\), \(i=1,\dots,m\) such that for \(\varepsilon>0\) small enough our problem (either Liouville or mean field form) has a solution \(u_\varepsilon\) blowing up positively around each \(\xi_1,\dots,\xi_{m_1}\) and negatively around \(\xi_{m_1+1},\dots,\xi_m\) respectively, as \(\varepsilon\to0\). We have used a family of solutions of the singular Liouville equation + \[ + \Delta u+|x -\xi|^{\alpha-2}e^{u}=0 \quad\text{ in $\text{I\!R}^2$, }\quad\text{satisfying}\qquad + \displaystyle\int_{\text{I\!R}^2} |x -\xi |^{\alpha-2} e^u <+\infty + \] + to construct an approximation of the solution depending on some parameters, suitable projected and scaled in order to make the error small enough in a suitable norm. Hence, we have found an actual solution as a small additive perturbation of this initial approximation by using a fixed point argument. + Part of these results have been obtained in collaboration with Pierpaolo Esposito (Università di Roma Tre, Italia) and Angela Pistoia (Università di Roma ``La Sapienza'', Italia). + start: + end: + speaker: Pablo Figueroa (Universidad Austral de Chile, Chile) + - titulo: Non Linear Mean Value Properties for Monge-Ampère Equations + abstract: | + In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized $p-$Laplacian discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game. + Our goal in this talk is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Amp\`ere equation. + Joint work with P. Blanc (Jyväskylä), F. Charro (Detroit), and J.J. Manfredi (Pittsburgh). + start: + end: + speaker: Julio D. Rossi (Universidad de Buenos Aires, Argentina) + - titulo: Radially symmetric solutions to quasilinear problems with indefinite weight + abstract: | + In this talk we present recent results regarding the existence of infinitely many radially symmetric solutions to + \[\label{eq main problem} + \left\{ + \begin{aligned} + - \Delta_p u =&W(| x|)g(u) &\hbox{in} &\quad \,\,\,B_1 (0)\subset \mathbb{R} ^N, \,\\ + u =& 0 & \hbox{in} & \quad \partial B_1 (0), + \end{aligned} + \right. + \] + where \(N\geq 2\), \(p>1\), \(\Delta _p u= \text{div} (|\nabla u|^{p-2} \nabla u)\) is the p-Laplacian operator, \(B_1 (0)\) stands for the unit ball in \(\mathbb{R} ^N\) centered at the origin, \(g: \mathbb{R} \longrightarrow \mathbb{R} \) is a nonlinear function which satisfies p-superlinearity conditions and the weight \(W: [0,1]\longrightarrow \mathbb{R}\) is a \(C^1 -\)function that changes sign. + Since the weight \(W\) takes negative values, the solutions to initial value problems associated to \eqref{eq main problem} may blow up at some \(r \in (X,1)\) and this fact represents a major difficulty when using phase-plane analysis arguments. We will discuss how this difficulty can be surpassed. + This is a recent work in collaboration with Alfonso Castro, Jorge Cossio and Sigifredo Herrón. + start: + end: + speaker: Carlos Vélez (Universidad Nacional de Colombia, Colombia) + - titulo: 'A nonlocal isoperimetric problem: density perimeter' + abstract: | + We will discuss a variant of a classical geometric minimization problem in arbitrary dimensions \(d\geq 2\), known as nonlocal isoperimetric problem, which arises from studies in Nuclear Physics by Gamow in the 1930’s: + \[ + \inf_{\substack{\Omega\subset\mathbb{R}^d\\|\Omega|=m}}E_{a}^{\gamma}(\Omega):=\int_{\partial^*\Omega}a(x)d\mathcal{H}^{d-1}+\gamma\iint_{\Omega\times\Omega}\frac{1}{|x-y|^{\alpha}}dxdy + \] + for any \(\alpha\in(0,d)\) and \(\gamma>0\). By introducing a density \(a:\mathbb{R}^d\to[0,\infty)\) in the perimeter functional, we obtain features that differ substantially from existing results in the framework of the classical problem without densities. Firstly, we get existence of minimizing sets for a general class of coercive densities \(a(x)\) and all values of ``mass". In addition, we prove essential boundedness of such minimizers. Finally, in the regime of “small” non-local contribution we completely characterize the minimizer, for densities that are monomial radial weights. This work is a collaboration with Stan Alama and Lia Bronsard (McMaster University) and Ihsan Topaloglu (Virginia Commonwealth University), as part of the project QUALITATIVE PROPERTIES OF WEIGHTED AND ANISOTROPIC VARIATIONAL PROBLEMS financed by ANID CHILE FONDECYT INICIACION No. 11201259. + start: + end: + speaker: Andres Zuñiga (Universidad de O’Higgins, Chile) - sesion: Stochastic processes and applications organizadores: - nombre: Joaquín Fontbona @@ -959,18 +1666,167 @@ mail: dremenik@dim.uchile.cl - nombre: Víctor Rivero mail: rivero@cimat.mx + charlas: + - titulo: Lattice trees in high dimensions + abstract: | + Lattice trees is a probabilistic model for random subtrees of \(\mathbb{Z}^d\). In this talk we are going to review some previous results about the convergence of lattice trees to the "Super-Brownian motion" in the high-dimensional setting. Then, we are going to show some new theorems which strengthen the topology of said convergence. Finally, if time permits, we will discuss the applications of these results to the study of random walks on lattice trees. + Joint work with A. Fribergh, M. Holmes and E. Perkins. + start: + end: + 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. + start: + end: + 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. + start: + end: + speaker: Octavio Arizmendi (Centro de Investigación en Matemáticas, México) + - titulo: PageRank Nibble on directed stochastic block models + abstract: | + This talk will focus on the probabilistic analysis of the PageRank algorithm on directed stochastic block models with K communities. Specifically, we show that for sparse graphs, i.e., with stochastically bounded degrees, the distribution of a randomly chosen vertex within a given community converges in a Wasserstein metric as the number of vertices grows to infinity. Moreover, the set of limiting distributions for typical vertices in each of the K communities can be characterized via a system of branching stochastic fixed-point equations (SFPEs). We then show how this characterization via SFPEs can be used to analyze the PageRank Nibble algorithm, which is a version of personalized PageRank that can be used for community detection provided one has a seed set of vertices known to belong to the community of interest. Our approach provides a good simple heuristic for choosing the optimal damping factor and provides computable bounds for the probability of misclassification. + start: + end: + speaker: Mariana Olvera-Cravioto (University of North Carolina at Chapel Hill, EEUU) + - titulo: Sharp bounds for consensus-based optimization of convex functions + abstract: | + Suppose one wants to optimize a function \(f:{\bf R}^d\to{\bf R}\). However, the function is not directly available, and all one can do is to use a ``black box" that on a selected input \(x\) produces the output \(f(x)\). Consensus-based optimization is a collection of techniques for optimizing \(f\) in this black box setting which is based on mean-field particle systems. We study a variant of this general method in the setting where \(f\) is strongly convex and smooth, and obtain sharp convergence bounds. This is work in progress with Dyego Araújo (IMPA). + start: + end: + speaker: Roberto Imbuzeiro Oliveira (Instituto de Matemática Pura e Aplicada, Brasil) + - titulo: Condensating zero-range process with condensation + abstract: | + Zero-range processes are interacting particle models where particles jump between different sites according to rates that only depend on the number of particles at the current position. In this talk we will focus on the case when the rates decrease with the number of particles, inducing a condensation phenomenon where a macroscopic number of particles occupies the same position. I will present recent results that identify the coarsening dynamics of the process as fluid limit. These techniques can be applied to derive fluid limits for Jackson networks with rates that depend on the number of customers in the queue. Joint work with Johel Beltrán, Daniela Cuesta, Milton Jara and Matthieu Jonckheere. + start: + end: + speaker: Inés Armendáriz (Universidad de Buenos Aires, Argentina) + - titulo: A martingale approach to lumpability + abstract: | + The martingale problem introduced by Stroock and Varadhan is an efficient method to prove the convergence of a sequence of stochastic processes which are derived from Markov processes. In this talk we present two examples to illustrate this approach: the density of particles per site of a sequence of condensing zero range processes and the number of sites occupied by a system of coalescing random walks evolving on a transitive finite graph. Both examples exhibit a sort of asymptotic lumpability. + start: + end: + speaker: Johel Beltran (Pontificia Universidad Católica del Perú, Perú) + - titulo: Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance + abstract: | + This talk presents recent results on the cutoff phenomenon for a general class of asymptotically exponentially stable Ornstein-Uhlenbeck systems under ε-small additive Lévy noise. The driving noise processes include Brownian motion, α-stable Lévy flights, finite intensity compound Poisson processes and red noises and may be highly degenerate. Window cutoff thermalization is shown under generic mild assumptions, that is, we see an asymptotically sharp ∞/0-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure με along a time window centered in a precise ε-dependent time scale tε . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of the drift matrix Q. With this piece of theory at hand we provide a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to ε-small Brownian motion or α-stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature. + Joint work with Gerardo Barrera (University of Helsinki, Finland) and Juan Carlos Pardo (CIMAT, México). + start: + end: + speaker: Michael Hoegele (Universidad de los Andes, Colombia) - sesion: Large Stochastic systems organizadores: - nombre: Claudio Landim mail: landim@impa.br - nombre: Pablo Ferrari mail: pferrari@dm.uba.ar + charlas: + - titulo: From stochastic dynamics to fractional PDEs with several boundary conditions + abstract: | + In this seminar I will describe the derivation of certain laws that rule the space-time evolution of the conserved quantities of stochastic processes. The random dynamics conserves a quantity (as the total mass) that has a non-trivial evolution in space and time. The goal is to describe the connection between the macroscopic (continuous) equations and the microscopic (discrete) system of random particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling; while the latter is a collection of particles that move randomly according to a transition probability and rings of Poisson processes. I will focus on a model for which we can obtain a collection of (fractional) reaction-diffusion equations given in terms of the regional fractional Laplacian with different types of boundary conditions. + start: + end: + speaker: Patrícia Gonçalves (Instituto Superior Técnico, Portugal) + - titulo: Persistence phenomena for large biological neural networks + abstract: | + We study a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of biological mechanisms. We focus here on local interactions: upon firing, a given neuron increases the potential of a fixed number of randomly chosen neurons. We show a phase transition in terms of the stationary distribution of the limiting network. Whereas a finite network activity always vanishes in finite time, the infinite network might converge to either a trivial stationary measure or to a nontrivial one. This allows to model the biological phenomena of persistence: we prove that the network may retain neural activity for large times depending on certain global parameters describing the intensity of interconnection. + Joint work with: Maximiliano Altamirano, Roberto Cortez, Lasse Leskela. + start: + end: + speaker: Matthieu Jonckheere (Universidad de Buenos Aires, Argentina) + - titulo: Structure recovery for partially observed discrete Markov random fields on graphs + abstract: | + Discrete Markov random fields on graphs, also known as graphical models in the statistical literature, have become popular in recent years due to their flexibility to capture conditional dependency relationships between variables. They have already been applied to many different problems in different fields such as Biology, Social Science, or Neuroscience. Graphical models are, in a sense, finite versions of general random fields or Gibbs distributions, classical models in stochastic processes. This talk will present the problem of estimating the interaction structure (conditional dependencies) between variables by a penalized pseudo-likelihood criterion. First, I will consider this criterion to estimate the interaction neighborhood of a single node, which will later be combined with the other estimated neighborhoods to obtain an estimator of the underlying graph. I will show some recent consistency results for the estimated neighborhood of a node and any finite sub-graph when the number of candidate nodes grows with the sample size. These results do not assume the usual positivity condition for the conditional probabilities of the model as it is usually assumed in the literature of Markov random fields. These results open new possibilities of extending these models to situations with sparsity, where many parameters of the model are null. I will also present some ongoing extensions of these results to processes satisfying mixing type conditions. This is joint work with Iara Frondana, Rodrigo Carvalho and Magno Severino. + start: + end: + speaker: Florencia Leonardi (Universidade de São Paulo, Brasil) + - titulo: An algorithm to solve optimal stopping problems for one-dimensional diffusions + abstract: | + Considering a real-valued diffusion, a real-valued reward function and a positive discount rate, we provide an algorithm to solve the optimal stopping problem consisting in finding the optimal expected discounted reward and the optimal stopping time at which it is attained. Our approach is based on Dynkin's characterization of the value function. The combination of Riesz's representation of r-excessive functions and the inversion formula gives the density of the representing measure, being only necessary to determine its support. This last task is accomplished through an algorithm. The proposed method always arrives to the solution, thus no verification is needed, giving, in particular, the shape of the stopping region. Generalizations to diffusions with atoms in the speed measure and to non smooth payoffs are analyzed. + start: + end: + speaker: Ernesto Mordecki (Universidad de la República, Uruguay) + - titulo: Branching processes with pairwise interactions + abstract: | + In this talk, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. BPI-processes form a subclass of branching processes with interactions, which were recently introduced by González Casanova et al. (2021), and includes the so-called logistic branching process which was studied by Lambert (2005). + Here, we provide a series of integral tests that fully explains how competition and cooperation regulates the long-term behaviour of BPI-processes. In particular, we give necessary and sufficient conditions for the events of explosion and extinction, as well as conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use the moment dual of BPI-processes which turns out to be a family of diffusions taking values on \([0,1]\), that we introduce as generalised Wright-Fisher diffusions together with a complete understanding of the nature of their boundaries. + start: + end: + speaker: Juan Carlos Pardo (Universidad Nacional Autónoma de México, México) + - titulo: Exact solution of TASEP and generalizations + abstract: | + I will present a general result which allows to express the multipoint distribution of the particle locations in the totally asymmetric exclusion process (TASEP) and several related processes, for general initial conditions, in terms of the Fredholm determinant of certain kernels involving the hitting time of a random walk to a curve defined by the initial data. This scheme generalizes an earlier result for the particular case of continuous time TASEP, which has been used to prove convergence of TASEP to the KPZ fixed point. The result covers processes in continuous and discrete time, with push and block dynamics, as well as some extensions to processes with memory length larger than 1. + Based on joint work with Konstantin Matetski. + start: + end: + speaker: Daniel Remenik (Universidad de Chile, Chile) + - titulo: Percolation on Randomly Stretched Lattices + abstract: | + In this talk we will give a new proof for a question that was first posed by Jonasson, Mossel and Peres, concerning percolation on a randomly stretched planar lattice. More specifically, we fix a parameter q in (0, 1) and we slash the lattice Z^2 in the following way. For every vertical line that crosses the x axis along an integer value, we toss an independent coin and with probability q we remove all edges along that line. Then we do the same with the horizontal lines that cross the y axis at integer values. We are then left with a graph G that looks like a randomly stretched version of Z^2 and on top of which we would like to perform i.i.d. Bernoulli percolation. The question at hand is whether this percolation features an non-trivial phase transition, or more precisely, whether p_c(G) < 1. Although this question has been previously solved in a seminal article by Hoffman, we present here an alternative solution that greatly simplifies the exposition. We also explain how the presented techniques can be used to prove the existence of a phase transition for other models with minimal changes to the proof. + This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis. + start: + end: + speaker: Augusto Teixeira (Instituto de Matemática Pura e Aplicada, Brasil) + - titulo: A martingale approach to lumpability + abstract: | + The martingale problem introduced by Stroock and Varadhan is an efficient method to prove the convergence of a sequence of stochastic processes which are derived from Markov processes. In this talk we present two examples to illustrate this approach: the density of particles per site of a sequence of condensing zero range processes and the number of sites occupied by a system of coalescing random walks evolving on a transitive finite graph. Both examples exhibit a sort of asymptotic lumpability. + start: + end: + speaker: Johel Beltrán (Pontificia Universidad Católica, Perú) - sesion: Geometría Algebraica computacional organizadores: - nombre: Gregorio Malajovich mail: gregorio.malajovich@gmail.com - nombre: Diego Armentano mail: diego@cmat.edu.uy + charlas: + - titulo: On the number of roots of random invariant homogeneous polynomials + abstract: | + The number of roots of random polynomials have been intensively studied for a long time. In the case of systems of polynomial equations the first important results can be traced back to the nineties when Kostlan, Shub and Smale computed the expectation of the number of roots of some random polynomial systems with invariant distributions. Nowadays this is a very active field. + In this talk we are concerned with the variance and the asymptotic distribution of the number of roots of invariant polynomial systems. As particular examples we consider Kostlan-Shub-Smale, random spherical harmonics and Real Fubini Study systems. + start: + end: + speaker: Federico Dalmao Artigas (Universidad de la República, Uruguay) + - titulo: Univariate Rational Sum of Squares + abstract: | + Landau in 1905 proved that every univariate polynomial with rational coefficients which is strictly positive on the reals is a sum of squares of rational polynomials. However, it is still not known whether univariate rational polynomials which are non-negative on all the reals, rather than strictly positive, are sums of squares of rational polynomials. + In this talk we consider the local counterpart of this problem, namely, we consider rational polynomials that are non-negative on the real roots of another non-zero rational polynomial. Parrilo in 2003 gave a simple construction that implies that if \(f\) in \(\mathbb R[x]\) is squarefree and \(g\) in \(\mathbb R[x]\) is non-negative on the real roots of \(f\) then \(g\) is a sum of squares of real polynomials modulo \(f\). Here, inspired by this construction, we prove that if \(g\) is a univariate rational polynomial which is non-negative on the real roots of a rational polynomial \(f\) (with some condition on \(f\) wrt \(g\) which includes squarefree polynomials) then it is a sum of squares of rational polynomials modulo \(f\). + Joint work with Bernard Mourrain (INRIA, Sophia Antipolis) and Agnes Szanto (North Carolina State University). + start: + end: + speaker: Teresa Krick (Universidad de Buenos Aires, Argentina) + - titulo: On eigenvalues of symmetric matrices with PSD principal submatrices + abstract: | + Real symmetric matrices of size n, whose all principal submatrices of size k|W_{f,\psi}(a,b_0\pm\epsilon)|.$$ + As usual, \(W_{f,\psi}\) stands for the CWT of the time series of the signal \(f\) using the mother wavelet \(\psi\) and \(a, b_0\) the dilation and translation parameters, respectively. This points of maximum modulus are then connected giving as resulta the maximum modulus lines. It can be shown that if a signal has a singularity at a given point, then there is a maximum modulus line that converges towards the location of the singularity at small scales. In smoothed singularities the maximum modulus line may end at a scale which is approximately the smoothing scale of the singularity. For this reason, the maximum modulus lines, which go from a fixed highest frequency/smallest scale to a smallest frequency/largest scale were considered and the estimated position of the singularity is simply the small-scale end of the corresponding maximum modulus line. + In our work we have considered as mother wavelet the following function, + $$\psi(t)=-\theta'(t), \qquad \theta(t)=\text{e}^{-t^2}.$$ + This mother wavelet has a null moment and the wavelet transform of the signal has a similar behaviour to the derivative of the signal. In this way, the CWT allows to detect the abrupt changes of the position, i.e. the singularities of its velocity. + Another method for detecting microsaccades is to consider them as outliers of the velocity since, as we have said before, they can be thought as ballistic movements. For the algorithm used in this method of detection, it is necessary to determine the value of a parameter \(\lambda\) and a minimum duration for the movement to be considered as a microsaccadic. Different works use different values for these parameters since it depends on the application and the determined value so that the detection is coherent. For our work we determined that \(\lambda=1.5\) and a minimum duration of 6 samples (i.e. 5ms duration) turn out to be a mean rate according to different works related to reading experiments. + Comparing the method of maximum modulus lines and the velocity algorithm using these parameters we have precisely determined the microsaccadic movements during the reading process. + For the experiment, we have considered two different groups of healthy people with similar education: one group of young adults and other group of elderly people. The first group consisted of 40 adults with mean age 28 (SD=4.2 years) and mean education 18.2 (in years). The second one involved 40 adults with mean age 71 (SD=6.1 years), mean education 15.1 (in years). We only took into account data of 19 subjects from the first group and 18 from the second one. To perform the experiment, we used a Spanish sentence corpus composed of 76 regular sentences that represent a large variety of grammatical structures, also called Low Predictability Sentences, and 64 Proverbs that are common our Argentinian culture. Each sentence was displayed in the centerline of a 20-inch LCD Monitor (1024\(\times\)768 pixels resolution; font: regular New Courier, 18 point, vertical size of one character: 0.2 in height). The participants were seated in front of the monitor at a distance of 60 cm. Head movements were minimized using a chin rest. The participants eye movements were recorded with an EyeLink 1000 Desktop Mount (SR Research) eyetracker, with a sampling rate of 1000 Hz and an eye position resolution of 20-s arc. All recordings and calibration were binocular. The participants gaze was calibrated with a standard 13-point grid for both eyes. After validating the calibration, a fixation point appeared at the position where the first letter of the sentence was to be presented. As soon as both eyes were detected within a one grade radius relative to the fixation point, the sentence was presented. After reading it, participants had to move their eyes to a dot in the lower right corner of the screen to end the trial. + The sentences ranged from 5 to 14 words in length; mean length being 7.3 (SD=1.9) words. The words ranged from 1 to 14 letters in length; the mean word length was 4.0 (SD=2) letters. All the participants read the whole set of sentences but the data with registration or reading errors were discarded. So the group of young participants read 3385 sentences and the group of adults read 2569 sentences. We have first detected the fixations on the words and then extracted the microsaccadic movements. For the first group were counted 27594 fixations with mean duration of 206.83ms (SD=105.58) while for the second one were counted 22571 fixations with a mean duration of 203.80ms (SD=100.30). + Now let us recall the definition of maxjumpword (MJW): it is the word with the largest difference between the close predictability of two consecutive words and can be determined according to the following equation + \[ maxjumpword=\text{max}[\text{logit}(pred_{N+1}-\text{logit}(pred_{N}], \] + being \(\text{logit}\big(pred_{N}\big)=0.5\hspace{2mm}\text{ln} \Big(\frac{pred_{N}}{1-pred_{N}}\Big)\) and \(pred_{N}\) the predictability of the read word \(N\). + In this work we extract the microsaccadics and perform a first analysis of changes in some of its characteristics respect to the relative position of the MJW. The considered characteristics are: length of the movement, vertical and horizontal variations, maximum velocity and duration. + start: + end: + speaker: Liliana R. Castro (Universidad Nacional del Sur, Argentina), joint with Juan M. Arriola and Marcela P. Álvarez + - titulo: Entropía y Complejidad Wavelet para identificar la inestabilidad eléctrica cardíaca en pacientes con infarto de miocardio + abstract: | + Se ha demostrado que aquellos individuos que han sufrido un infarto de miocardio (IM) tienen una alta probabilidad de desarrollar arritmias ventriculares malignas y/o muerte súbita cardíaca. Las anomalías de la conducción eléctrica que aparecen en la región infartada del miocardio se reflejan en el electrocardiograma (ECG) como fragmentaciones del complejo QRS (fQRS). Dichas fragmentaciones no siempre son posibles de detectar visualmente. Si bien puede suceder que los pacientes no desarrollen taquicardia y/o fibrilación venticular (VT/VF), son potencialmente riesgosos porque pueden desarrollarla inesperadamente y sin síntomas previos. + Hay pocas técnicas no invasivas para capturar dichas inestabilidades eléctricas, como la técnica de electrocardiograma de señal promediada, y la varianza espectral del complejo QRS, entre otras. + Por ello, hemos evaluado la Entropía y Complejidad Wavelet del complejo QRS del ECG, utilizando la Transformada Wavelet Continua, como un método eficaz para cuantificar estas alteraciones anormales en la actividad eléctrica cardíaca en pacientes post IM que no presentan VT/VF. + start: + end: + speaker: Gisela Clemente, (Universidad Nacional de La Plata, Argentina), joint with Esteban Valverde + - titulo: 'Wavelet Analysis in Space Research at the Applied Mathematics and Geophysics group at INPE: advances and overview' + abstract: | + This presentation aims at a comprehensive view of ongoing research efforts in multiscale analysis applied to challenges in the context of space environment and its electrodynamical effects on the Earth and nearby regions. Based on the developments by the INPE group LANCE and collaborators, the content explored takes an overview and the next steps on the multiscale research. We discussed some results obtained in the wavelet analysis of space and geophysical dataset and their implication on understanding the physics processes. Moreover, we present the first successful results obtained in the wavelet-based multiresolution regularity detection methodology with the generic block-structured mesh adaptation in evolutionary partial differential equations. In particular, the solvers that will permit fast and reliable space weather magneto-hydrodynamic simulations and their needs in the multiscale real time-space data modelling in future. + start: + end: + speaker: Margarete Oliveira Domingues (Instituto Nacional de Pesquisas Espaciais, Brasil), joint with Odim Mendes and Muller Moreira Lopes + - titulo: 'Cryopreservation Process: analyzed through Wavelet Transforms' + abstract: | + Rapid and ultra rapid cooling methods, in which liquid nitrogen (\(LN_2\)) is used as cooling agent, require special equipment and methods in order to record and analyze the thermal history of the sample to be cryopreserved. The temperature profile during the process of cooling is an important parameter in the design and description of any protocol for cryopreservation. We present a home made device, used to record the temperature at high rate during the cooling in \(LN_2\). System noise at room temperature is about \(\pm ~ 2.9 ^{\circ}\)C and at \(LN_2\) is about \(\pm ~ 6.9 ^{\circ}\)C, when in the liquid nitrogen it is (\(-196^{\circ}\)C). This noise introduced when measuring the temperature may, in some cases, difficult the analysis and interpretation of the signal. A related use of Wavelet transform is for smoothing and or de noising data based on wavelet coefficients, by means of removing the undesired frequency components. With the intention of eliminate high frequencies noise, we apply a filter in such a way that in the level ``\(j\)'' of the process, the residuals will be a soften version of the original signal having less amount of high frequency signals in contrast with level ``\(j+1\)'', and half number of data. These are the main points by which, in the present work, we have used a signal isolation method based on orthogonal wavelets, in order to analyze the remaining signal with minimum modification of the associated dynamics. Using this filtering method, system noise was virtually eliminated allowing a more precise interpretation of the significance of cooling curves. + start: + end: + speaker: Ana Korol (Universidad Nacional de Rosario, Argentina) + - titulo: Aplicación de Transformada Wavelet Continua a señales de Emisión Acústica de fractura de materiales frágiles + abstract: | + La Emisión Acústica es un ensayo no destructivo que nos permite monitorear la integridad de un material frágil sometido a esfuerzos. Si, por ejemplo, un cilindro de roca es comprimido hasta su fractura se generan ondas elásticas en el material que a través de un sensor se convierten en ondas eléctricas, denominadas señales de Emisión Acústica. Estas señales nos dan información sobre los procesos complejos que se producen en el material tales como nucleación, crecimiento y coalescencia de microfacturas hasta que finalmente se producen macrofracturas que llevan a la ruptura del material. Las señales explosivas de EA que se generan durante el proceso involucran un amplio rango de tiempos y escalas por lo que a través del análisis con TWC nos permite identificar y evaluar los diferentes procesos. + start: + end: + speaker: Rosa Piotrkowski (Universidad de Buenos Aires, Argentina), joint with Miguel Eduardo Zitto + - titulo: Uso de la Transformada Wavelet en el análisis de datos de la evolución del Covid-19 + abstract: | + Desde el comienzo de la pandemia originada por el SARS-COV 2, se han utilizado distintas herramientas para analizar los datos disponibles, a fin de comprender su dinámica. En gran medida, en el análisis, se hizo uso de modelos matemáticos diseñados para describir la evolución de la pandemia. Se aplicaron, además, técnicas de análisis de series temporales, las cuales tienen la ventaja de acumular información, al contar con un mayor volumen de datos con el paso del tiempo. El número de infectados y muertos diarios, entre otros datos, corresponden a series no estacionarias, dado que sus propiedades estadísticas varían con el tiempo. A modo de ejemplo, la cantidad de personas a infectarse tiende a reducirse a medida que más personas se infectan. Además, si el número de infectados aumenta, suelen tomarse diferentes medidas, tales como las que restringen la movilidad, al mismo tiempo que el avance de la vacunación tiende a producir una disminución de los casos. Todos estos factores, tomados de conjunto, le otorgan una dinámica compleja a las series a estudiar: en ellas conviven múltiples fenómenos localizados tanto en el tiempo como en frecuencia, que se superponen bajo complejas estructuras. La Transformada Wavelet se presenta como una herramienta apropiada para analizar series de este tipo, dado que puede ser aplicada a series no estacionarias, y permite estudiar las relaciones subyacentes entre sus datos en el tiempo y en frecuencia, así como encontrar patrones de comportamiento. En este trabajo aplicamos estas herramientas a las series de datos de infectados, fallecidos y de movilidad de la provincia de Buenos Aires y mostramos, apelando a distintos cuantificadores, tales como la coherencia wavelet, el análisis tiempo-frecuencia de las relaciones entre distintas series. De esta forma la Transformada Wavelet se coloca como una herramienta de análisis relevante a tener en cuenta en el proceso de toma de decisiones. + start: + end: + speaker: Victoria Vampa (Universidad Nacional de La Plata, Argentina), joint with Federico Holik - sesion: Geometría Algebraica organizadores: - nombre: Álvaro Rittatore mail: alvaro@cmat.edu.uy - nombre: Pedro Luis del Ángel R. mail: luis@cimat.mx + charlas: + - titulo: Birational geometry of Calabi-Yau pairs + abstract: | + Recently, Oguiso addressed the following question, attributed to Gizatullin: ``Which automorphisms of a smooth quartic K3 surface \(D\subset \mathbb{P}^3\) are induced by Cremona transformations of the ambient space \(\mathbb{P}^3\)?'' When \(D\subset \mathbb{P}^3\) is a quartic surface, \((\mathbb{P}^3,D)\) is an example of a \emph{Calabi-Yau pair}, that is, a pair \((X,D)\) consisting of a normal projective variety \(X\) and an effective Weil divisor \(D\) on \(X\) such that \(K_X+D\sim 0\). In this talk, I will explain a general framework to study the birational geometry of mildly singular Calabi-Yau pairs. This is a joint work with Alessio Corti and Alex Massarenti. + start: + end: + speaker: Carolina Araujo (Instituto de Matemática Pura e Aplicada, Brasil) + - titulo: On reconstructing subvarieties from their periods + abstract: | + We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. This is a joint work with Emre Sertoz. + start: + end: + speaker: Hossein Movasati (Instituto de Matemática Pura e Aplicada, Brasil) + - titulo: 'Quantum cohomology and derived categories: the case of isotropic Grassmannians' + abstract: | + The relation between quantum cohomology and derived categories has been known since long ago. In 1998 Dubrovin formulated his celebrated conjecture relating semisimplicity of the quantum cohomology of Fano manifolds and the existence of exceptional collections in their derived categories of coherent sheaves. In this talk we will discuss recent joint work of the author with Anton Mellit, Nicolas Perrin and Maxim Smirnov studying the relation between quantum cohomology for certain isotropic Grassmannians and their quantum cohomology in the spirit of Dubrovin conjecture. In this case the small quantum cohomology turns out not to be semisimple but still an interesting decomposition of the derived category arises. If time permits new directions and some problems will be mentioned. + start: + end: + speaker: John Alexander Cruz Morales (Universidad Nacional de Colombia, Colombia) + - titulo: Initial degeneration in differential algebraic geometry + abstract: | + By a degeneration, we mean a process that transforms a geometric object \(X/F\) defined over a field \(F\) into a simpler object that retains many of the relevant properties of \(X\). Formally, any degeneration is realized by an integral model for \(X\); that is, a flat scheme \(X'/R\) defined over some integral domain \(R\) whose generic fiber is the original object \(X\). + In this talk, we endow the field \(F=K((t_1,\ldots,t_m))\) of quotients of multivariate formal power series with a (generalized) non-Archimedean absolute value \(|\cdot|\). We use this to establish the existence of (affine) integral models \(X'/R\) over the ring of integers \(R=\{|x|\leq 1\}\) for schemes \(X\) associated to solutions of systems of algebraic partial differential equations with coefficients on \(F\). We also concretely describe the specialization map of a model \(X'/R\) to the maximal ideals of \(R\), which are encoded in terms of usual (total) monomial orderings. + start: + end: + speaker: Cristhian Garay López (Centro de Investigación en Matemáticas, México) + - titulo: Rank two bundles over fibered surfaces + abstract: | + Let S be a smooth complex surface fibered over a curve C. Under suitable assumptions, if E is a stable rank two bundle on C then its pullback is a H-stable bundle on S. In this sense we can relate moduli spaces of rank two stable bundles on the surface S with moduli spaces of rank two stable bundles on C. In this talk we aim to take advantage of this relation to provide examples of Brill–Noether loci on fibered surfaces. + start: + end: + speaker: Graciela Reyes (Universidad Autónoma de Zacatecas, México) + - titulo: Langlands Program and Ramanujan Conjecture + abstract: | + We will give an overview of several aspects of the Langlands Program that interconnect different fields of mathematics, namely: Algebraic Geometry, Number Theory and Representation Theory. We study general results over global fields, and mention what is known on Langlands functoriality conjectures. In characteristic p, we have a rich interplay with algebraic geometry and we present results of the author in connection with the work of Laurent and Vincent Lafforgue. As a main application, we look at the Ramanujan conjecture, known for generic representations of the classical groups and GL(n) over function fields, for example. + start: + end: + speaker: Luis Alberto Lomelí (Pontificia Universidad Católica, Chile) + - titulo: Actions and Symmetries + abstract: | + I will discuss some classical and some recent results about group and algebra actions on abelian varieties and curves. + start: + end: + speaker: Rubí Rodríguez (Universidad de la Frontera, Chile) + - titulo: Representation of groups schemes that are affine over an abelian variety + abstract: | + The classical representation theory of affine schemes over a field, has been widely considered and many of its main properties have been understood and put under contral--after the work of many mathematicians along the second half of the 20th century. In joint work with del Ángel and Rittatore, we proposed a representation theory for the more general situation where the group scheme is affine over a fixed abelian variety A (the classical case corresponds to the situation that A is the trivial abelian variety with only one point). In this talk we describe the main relevant definitions, where the representations are vector bundles over A with suitable actions of the group scheme G. Then, we show that in our context an adequate formulation of Tannaka--type of recognition and reconstruction results (due to Grothendieck, Saavedra, Deligne and others) remain valid. + start: + end: + speaker: Walter Ferrer (Universidad de la República, Uruguay) + - titulo: A criterion for an abelian variety to be non-simple and applications + abstract: | + 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 \(g\). + 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 + We developed this criterion to apply it in the problem of finding completely decomposable Jacobian varieties. This research is motivated by Ekedahl and Serre's questions in Ekedahl, Serre (1993), which can be summarized into whether there are curves of arbitrary genus with completely decomposable Jacobian variety. Moreover, in Moonen, Oort (2004), the authors ask about the existence of positive dimensional special subvarieties \(Z\) of the Jacobian loci \(\mathcal{T}_g\) in the moduli space of principally polarized abelian varieties, such that the abelian variety corresponding with the geometric generic point of \(Z\) is isogenous to a product of elliptic curves. + In this direction, we use this criterion in combination with the so called {\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 + \[B_1^{n_1} \times \cdots \times B_r^{n_r}\to JX\] + Although \(JX\) is principally polarized, the induced polarization on \(B_i\) is in general not principal. The question we address is to study the simplicity of \(B_i\), hence finding in some cases a further decomposition of \(JX\). + This is a joint work with R. Auffarth and H. Lange. + start: + end: + speaker: Anita Rojas (Universidad de Chile, Chile) - sesion: K-teoría organizadores: - nombre: Noé Bárcenas mail: barcenas@matmor.unam.mx - nombre: Eugenia Ellis - mail: eellis@fing.edu.uy + mail: eellis@fing.edu.uy + charlas: + - titulo: K-teoría y problemas de clasificación + abstract: | + El formato general de los problemas de clasificación que nos ocupan es el siguiente. Se tienen un anillo de base \(\ell\), una familia \(C\) de \(\ell\)-álgebras, y un invariante \(K\)-teórico (e.g. \(K_0\) con su preorden natural), y nos preguntamos si dos integrantes de la familia con invariantes isomorfos son necesariamente isomorfas o equivalentes bajo algún criterio (e.g. Morita equivalentes). Una fuente de estos problemas es tomar una clase de \(C^*\)-álgebras (e.g. completaciones de \(\mathbb{C}\)-álgebras definidas por cierto tipo de generadores y relaciones) que ha sido clasificada por su \(K\)-teoria (topológica) y preguntarse si la \(K\)-teoría algebraica clasifica el análogo puramente algebraico de esa familia (e.g. la dada por el mismo tipo de generadores y relaciones, sin completar). En la charla veremos algunos ejemplos de problemas de ese tipo y algunos resultados. + start: + end: + speaker: Guillermo Cortiñas (Universidad de Buenos Aires, Argentina) + - titulo: Algebraic K theory of 3-manifold groups + abstract: | + We describe the algebraic K-theory groups \(K_i(Z[G[))\) where G is the fundamental group of a closed compact 3-manifold and all \(i\in Z\). + start: + end: + speaker: Daniel Juan (Universidad Nacional Autónoma de México, México) + - titulo: Clasificación homotópica de álgebras de Leavitt simples puramente infinitas + abstract: | + En esta charla hablaremos del problema de clasificación de álgebras de Leavitt simples puramente infinitas sobre un grafo finito \(E\) y un cuerpo \(\ell\). Dado un grafo \(E\) tenemos asociado una \(\ell\)-álgebra de Leavitt \(L(E)\). Es un problema abierto decidir cuando el par \((K_0(E), [1_{L(E)}])\), que consiste del grupo de Grothendieck junto con la clase \([1_{L(E)}]\) de la identidad es un invariante completo, a menos de isomorfismo, para la clasificación de álgebras de Leavitt simples puramente infintas de grafo finito. Mostraremos que el par \((K_0(E), [1_{L(E)}])\) es un invariante completo para la clasificación a menos de equivalencia homotópica polinomial. Para probar esto vamos a estudiar la \(K\)-teoría álgebraica bivariante entre álgebras de Leavitt. + start: + end: + speaker: Diego Montero (Despegar, Argentina) + - titulo: Quasi-isometric Rigidity of subgroups and filtered ends + abstract: | + Let G and H be quasi-isometric finitely generated groups and let \(P\leq G\); is there a subgroup Q (or a collection of subgroups) of H whose left cosets coarsely reflect the geometry of the left cosets of P in G? We explore sufficient conditions for a positive answer. + start: + end: + speaker: Luis Jorge Sanchez Saldaña (Universidad Nacional Autónoma de México, México) + - titulo: A Gillet-Waldhausen Theorem for chain complexes of sets + abstract: | + In recent work, Campbell and Zakharevich introduced a new type of structure, called ACGW-category. These are double categories satisfying a list of axioms that seek to extract the properties of abelian categories which make them so particularly well-suited for algebraic K-theory. The main appeal of these double categories is that they generalize the structure of exact sequences in abelian categories to non-additive settings such as finite sets and reduced schemes, thus showing how finite sets and schemes behave like the objects of an exact category for the purpose of algebraic K-theory. + In this talk, we will explore the key features of ACGW categories and the intuition behind them. Then, we will move on to ongoing work with Brandon Shapiro where we further develop this program by defining chain complexes and quasi-isomorphisms for finite sets. These satisfy an analogue of the classical Gillet--Waldhausen Theorem, providing an alternate model for the K-theory of finite sets. + start: + end: + speaker: Maru Sarazola (Johns Hopkins University, Estados Unidos) + - titulo: Homología de Hochschild topológica y métodos de traza + abstract: | + Para un anillo \(R\), o más en general un espectro en anillos ("ring spectrum"), Quillen construyó la K-teoría algebraica superior de \(R\) como un espectro en anillos \(K(R)\). Se puede construir otro espectro \(THH(R)\), la homología de Hochschild topológica de \(R\): es un análogo de la homología de Hochschild clásica, solo que en vez de ser sobre el anillo de los enteros \(\mathbb{Z}\), es sobre una base más profunda, el espectro en anillos \(S\), llamado "espectro de las esferas" ("sphere spectrum"). La relación está dada por un mapa \(K(R) \rightarrow THH(R)\) llamado traza, o traza de Dennis topológica, que a veces permite acercarse al cálculo de \(K(R)\). Esbozaremos estas construcciones y daremos un ejemplo de cálculo. + start: + end: + speaker: Bruno Stonek (Instituto de Matemáticas de la Academia Polaca de Ciencias, Polonia) + - titulo: Topología controlada y conjeturas de isomorfismo en \(K\)-teoría + abstract: | + Sean \(R\) un anillo, \(G\) un grupo, \(\mathcal{F}\) una familia de subgrupos y \(E_{\mathcal{F}}G\) el \(G\)-\(CW\)-complejo universal con isotropía en \(\mathcal{F}\). La conjetura de isomorfismo identifica, mediante un morfismo de ensamble, la \(K\)-teoría del anillo de grupo \(\mathbf{K}(RG)\) con una teoría de homología equivariante evaluada en \(E_{\mathcal{F}}G\). Una construcción de esta teoría de homología se puede hacer utilizando topología controlada. En este contexto la conjetura afirma que el morfismo de ensamble + \[\partial_{\mathcal{F}_n}: K_{n+1}(\mathcal{D}^G(E_{\mathcal{F}}G))\to K_n(RG)\] + es un isomorfismo. Para \(\mathcal{F}=\mathcal{V}cyc\) la familia de subgrupos virtualmente cíclicos, la conjetura se conoce como \emph{conjetura de Farrell-Jones} y ha sido probada para una gran clase de grupos. + En este trabajo consideramos los casos en que \(G=\langle t\rangle\) es el grupo cíclico infinito y \(\mathcal{F}\) es la familia trivial, y \(G=D_{\infty}\) es el grupo diedral infinito con la familia de subgrupos finitos. En ambos casos \(\mathbb{R}\) es un modelo para \(E_{\mathcal{F}}G\) y su métrica nos permite dar una noción de tamaño a los morfismos de \(RG\)-módulos. Mostraremos c\'omo utilizar técnicas de control para analizar la suryectividad del morfismo de ensamble \(\partial_{\mathcal{F}_1}\). En el caso en que \(G=\langle t\rangle\) el morfismo de ensamble se identifica con el morfismo de Bass-Heller-Swan + \[K_0(R)\oplus K_1(R)\to K_1(R[t^{-1},t]),\] + que resulta un isomorfismo para R regular. (Trabajo en conjunto con E. Ellis, E. Rodríguez Cirone y S. Vega.) + start: + end: + speaker: Gisela Tartaglia (Universidad Nacional de La Plata, Argentina) + - titulo: Proper actions and equivariant K-theory of group extensions + abstract: | + In this talk we present some decompositions appearing in equivariant K-theory. Let \[0\to A\to G\to Q\to0\] be an extension of Lie groups with \(A\) compact, let \(X\) be proper \(Q\)-space, we present a decomposition of the \(G\)-equivariant K-theory of \(X\) in terms of twisted equivariant K-theories of \(X\) respect to certain subgroups of \(Q\). We present some examples of computations using this decomposition. Finally we give some ideas of how to extend this decomposition in the context of C*-algebras. + start: + end: + speaker: Mario Velazquez (Universität Göttingen, Alemania) - sesion: Stochastic analysis and stochastic processes organizadores: - nombre: Paavo Salminen @@ -999,6 +2063,59 @@ mail: Antoine.Lejay@univ-lorraine.fr - nombre: Ernesto Mordecki mail: mordecki@cmat.edu.uy + charlas: + - titulo: Optimal couplings for interacting particle systems via optimal transport + abstract: | + We will review some ideas recently developed in order to obtain optimal convergence rates for mean-field stochastic particle systems interacting through binary jumps, by constructing trajectorial couplings that make use of optimal transport plans for empirical measures. Examples of applications in kinetic theory and in branching population models will be discussed. + Based on joint works with Roberto Cortez and Felipe Muñoz. + start: + end: + speaker: Joaquin Fontbona (Universidad de Chile, Chile) + - titulo: Estimating the parameter of the Skew Brownian motion + abstract: | + The Skew Brownian motion may be constructed by choosing randomly the sign of a Brownian excursion. It thus depends on a parameter \(\theta\in[-1,1]\). In this talk, we discuss the statistical properties of the maximum likelihood estimator (MLE) for this parameter \(\theta\). In particular, it is consistent and asymptotically mixed normal as the local time of the process is involved in the limit, yet with a rate \(1/4\) and not \(1/2\) as usual. We also discuss some non-asymptotic properties of the MLE. + Based on joint work with Sara Mazzonetto (IECL, Université de Lorraine, Nancy). + start: + end: + speaker: Antoine Lejay (Institut Élie Cartan de Lorraine, Francia) + - titulo: Diffusion through a two-sided membrane + abstract: | + We study the Markov chains on \(\mathbb{Z}^d\), \(d\geq 2\), that behave like a standard symmetric random walk outside of the hyperplane (membrane) \(H=\{0\}\times \mathbb{Z}^{d-1}\). The transition probabilities on the membrane \(H\) are periodic and also depend on the incoming direction to \(H\), that makes the membrane \(H\) two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a \(d\)-dimensional diffusion whose first coordinate is a skew Brownian motion and the other \(d-1\) coordinates is a Brownian motion with a singular drift controlled by the local time of the first coordinate at \(0\). In the proof we utilize a novel martingale characterization of the Walsh Brownian motion and determine the effective permeability and slide direction. Eventually, a similar convergence theorem is established for the one-sided membrane without slides and random iid transition probabilities. + Based on joint work with A. Pilipenko (Institute of Mathematics, NAS of Ukraine, Kiev). + start: + end: + speaker: Ilya Pavlyukevich (Universität Jena, Alemania) + - titulo: Non-zero-sum optimal stopping game with continuous versus periodic observations + abstract: | + We introduce a new non-zero-sum game of optimal stopping with asymmetric information. Given a stochastic process modelling the value of an asset, one player has full access to the information and observes the process completely, while the other player can access it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a Lévy process with positive jumps, we not only prove the existence, but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. + Based on joint work with Kazutoshi Yamazaki and Neofytos Rodosthenous. + start: + end: + speaker: José Luis Pérez Garmendia (Centro de Investigación en Matemáticas, México) + - titulo: 'Diffusion spiders: resolvents, excessive functions and optimal stopping' + abstract: | + Let \(\Gamma\) be a graph with one internal node \(\mathbf{0}\) and \(n\geq 1\) edges meeting each other at \(\mathbf{0}\). For \(x>0\) and \(i\in\{1,2,\dots,n\}\) we let \(\mathbf{x}=(x,i)\) denote the point on \(\Gamma\) located on the \(i\)'th edge at the distance \(x\) from \(\mathbf{0}\). + Let \(X=(X_t)_{t\geq 0}\) be a one-dimensional diffusion on \(\mathbf{R}_+\) hitting 0 with probability one. A (homogeneous) diffusion spider \(\mathbf{X}=(\mathbf{X}_t)_{t\geq 0}\) is a continuous strong Markov process on \(\Gamma\) which on each edge before hitting \(\mathbf{0}\) behaves as \(X\) before hitting 0. When reaching \(\mathbf{0}\) the spider chooses, roughly speaking, an edge with some given probability to continue the movement. A rigorous construction of \(\mathbf{X}\) can be done using the excursion theory. + In this talk we give an explicit expression for the resolvent kernel of \(\mathbf{X}\). Using this we study the excessive functions of \(\mathbf{X}\) and derive the so called glueing condition to be satisfied at \(\mathbf{0}\). We apply the representation theory (Martin boundary theory) of excessive functions and calculate the representing measure of a given excessive function. + This machinery can be used to solve optimal stopping problems for \(\mathbf{X}\). As an example we consider Walsh's Brownian spider where the diffusion \(X\) is a Brownian motion. We focus on the case where the reward is continuous at \(\mathbf{0}\), linear on the edges and such that the resulting stopping region is connected. + Based on joint work with Jukka Lempa and Ernesto Mordecki. + start: + end: + speaker: Paavo Salminen (Åbo Akademi University, Finlandia) + - titulo: On the consistency of the least squares estimator in models sampled at random times driven by long memory noise + abstract: | + In numerous applications data are observed at random times. Our main purpose is to study a model observed at random times incorporating a long memory noise process with a fractional Brownian Hurst exponent \(H\). In this talk, we propose a least squares (LS) estimator in a linear regression model with long memory noise and a random sampling time. The strong consistency of the estimator is established, with a convergence rate depending on \(N\) and Hurst exponent. A Monte Carlo analysis supports the relevance of the theory and produces additional insights. + start: + end: + speaker: Soledad Torres (Universidad de Valparaiso, Chile) + - titulo: Sufficient variability of paths and differential equations with BV-coefficients + abstract: | + Partial differential equations and differential systems play a fundamental role in many aspects of our daily life. However, in many applications, especially in the field of stochastic differential or partial differential equations, the underlying equation does not make sense in a classical way, and one has to consider integral equations instead. Moreover, even the concept of integral is subtle. For example, a typical situation is that one needs to consider \(\int_0^t \varphi(X_s)\,dY_s\), where \(\varphi\) is a given function and \(X,Y\) are some continuous but non-differentiable objects. Several powerful techniques such as rough path theory have emerged to treat these situations, and it can be safely stated that nowadays differential systems driven by rough signals are already rather well understood. The idea in the rough path theory is, roughly speaking, that one assumes additional smoothness on the function \(\varphi\) in order to compensate bad behaviour of signal functions \(X\) and \(Y\). As such, the methodology cannot be applied in any straightforward manner if one allows discontinuities in \(\varphi\). In particular, this is the case if \(\varphi\) is a general BV-function. + In this talk we combine tools from fractional calculus and harmonic analysis to some fine properties of BV-functions and maximal functions, allowing us to give a meaningful definition for (multidimensional) integrals \(\int_0^t \varphi(X_s)\,dY_s\) with a BV-function \(\varphi\), provided that the functions \(X\) and \(Y\) are regular enough in the Hölder sense. Here enough regularity means better Hölder regularity than what is customary assumed in the rough path theory, and this gain in regularity can be used to compensate ill behaviour of \(\varphi\). The key idea is that the signal \(X\) should not spend too much time, in some sense, on the bad regions of \(\varphi\). We quantify this in terms of potential theory and Riesz energies. We also discuss several consequences, and provide existence and uniqueness results for certain differential systems involving BV-coefficients. Extensions and further topics are discussed. + Based on joint work with Michael Hinz (Bielefeld University) and Jonas Tölle (University of Helsinki). + start: + end: + speaker: Lauri Viitasaari (Aalto University, Finlandia) - sesion: Control and Stabilization for Partial Differential Equations organizadores: - nombre: Fágner D. Araruna @@ -1007,6 +2124,89 @@ mail: eduardo.cerpa@usm.cl - nombre: Luz de Teresa mail: deteresa@matem.unam.mx + charlas: + - titulo: Small-time global exact controllability to the trajectories of the Boussinesq system + abstract: | + In this talk, we consider the global exact controllability problem to the trajectories of the Boussinesq system. We show that it is possible to drive the solution to the prescribed trajectory in small time by acting on the system through the velocity and the temperature on an arbitrary small part of the boundary. The proof relies on three main arguments. First, we transform the problem into a distributed controllability problem by using a domain extension procedure. Then, we prove a global approximate controllability result by following a strategy of Coron and collaborators, which deals with the Navier-Stokes equations. This part relies on the controllability of the inviscid Boussinesq system and asymptotic boundary layer expansions. Finally, we conclude with a local controllability result that we establish with the help of a linearization argument and appropriate Carleman estimates. + start: + end: + speaker: Felipe Chaves (Universidade Federal de Paraiba, Brasil) + - titulo: 'Identification of a boundary obstacle in a Stokes fluid with Navier–slip boundary conditions: an exterior approach' + abstract: | + The problem of identifying an obstruction into a fluid duct has several major applications, for example in medicine the presence of a stenosis in a coronary vessels is a life threaten disease. In this talk we formulate a continuous setting and study from a numerical perspective the inverse problem of identifying an obstruction contained in a 2D elastic duct where a Stokes flow becomes turbulent after hitting the boundary (Navier--slip boundary conditions), generating an acoustic waves. To be precise, by using acoustic wave measurements at certain points at the exterior to the duct, we are able to identify the location; extension and height of the obstruction. Thus, our framework constitutes an external approach for solving this obstacle inverse problem. Synthetic examples are used in order to verify the effectiveness of the proposed numerical formulation. + In collaboration with: L. Breton, J. López Estrada and C. Montoya + start: + end: + speaker: Pedro González Casanova (Universidad Nacional Autónoma de México, México) + - titulo: Controllability from the exterior of fractional heat equation + abstract: | + In this talk, we consider the controllability problem from the exterior for the one dimensional heat equation on the interval \((-1,1)\) associated with the fractional Laplace operator \((-\partial_x^2)^s\), where \(00\) if and only if \(\frac 120\). We are interested in the case of a single control \(h\) acting on the exterior part of the boundary, with a given boundary condition imposed on the interior boundary: + $$ + \left\{ + \begin{array}{ll} + \partial_{tt} y - \Delta y = 0, & + \hbox{ in } (0,T) \times \Omega, \\ + {\mathcal B}y = 0, & (0,T) \times \partial B_{R_0}. \\ + y(t,x) = {h(t,x)}, & \hbox{ on } (0,T) + \times \partial B_{R_1}. + \end{array} \right. + $$ + We provide a robust strategy to prove the observability of this system with any of several boundary conditions on the internal boundary \((0,T) \times \partial B_{R_0}\), as Fourier conditions, dynamic conditions, conditions with the fractional Laplacian or fluid-structure models. + The main tool we use is given by resolvent characterizations, and the use of adequate estimates. We prove observability estimates with observations performed on the whole external boundary, which are valid for all the mentioned cases for boundary conditions at \(\partial B_{R_0}\). We will mention also recent related results concerning wave and Schrodinger equations in more general domains with the same topological structure. + start: + end: + speaker: Alberto Mercado (Universidad Técnica Federico Santa María, Chile) + - titulo: Internal controllability of the Kadomtsev-Petviashvili II equatio + abstract: | + In this talk, we present the internal control problem for the Kadomstev-Petviashvili II equation, better known as KP-II. The problem is studied first when the equation is set in a vertical strip proving by the Hilbert Unique Method and semiclassical techniques proving the internal controllability and second, in a horizontal strip where the controllability in $L^2(\T)$ cannot be reached. + start: + end: + speaker: Ivonne Rivas (Universidad del Valle, Colombia), joint with Chenmin Sun - sesion: Geometría Diferencial organizadores: - nombre: Romina Arroyo @@ -1015,9 +2215,124 @@ mail: delbarc@ime.unicamp.br - nombre: Silvio Reggiani mail: reggiani@fceia.unr.edu.ar + charlas: + - titulo: Pluriclosed metrics and Kähler-like conditions on complex manifolds + abstract: | + A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) or pluriclosed if the torsion of the associated Bismut connection associated is closed. I will present some general results on pluriclosed metrics in relation to symplectic geometry, the pluriclosed flow and Kähler-like curvature conditions. + start: + end: + speaker: Anna Fino (Università di Torino, Italia) + - titulo: The prescribed cross curvature problem + abstract: | + Chow and Hamilton introduced the notion of cross curvature and the associated geometric flow in 2004. Several authors have built on their work to study the uniformisation of negatively curved manifolds, Dehn fillings, and other topics. Hamilton conjectured that it is always possible to find a metric with given positive cross curvature on the three-sphere and that such a metric is unique. We will discuss several results that support the existence portion of this conjecture. Next, we will produce a counterexample showing that uniqueness fails in general. + start: + end: + speaker: Artem Pulemotov (The University of Queensland, Australia), joint with Timothy Buttsworth (The University of Queensland) + - titulo: Soliton solutions to the curve shortening flow on the 2-dimensional hyperbolic space + abstract: | + We prove that a curve is a soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space if and only if its geodesic curvature is given as the inner product between its tangent vector field and a vector of the 3-dimensional Minkowski space. We prove that there are three classes of such solutions and for each fixed vector there exits a 2-parameter family of soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space. Moreover, we prove that each soliton is defined on the whole real line, it is embedded and its geodesic curvature, at each end, converges to a constant + start: + end: + speaker: Keti Tenenblat (Universidade de Brasília, Brasil), joint with Fabio Nunes da Silva (Universidade Federal do Oeste da Bahia) + - titulo: Upper bound on the revised first Betti number and torus stability for RCD spaces + abstract: | + Gromov and Gallot showed that for a fixed dimension \(n\) there exists a number \(\varepsilon(n)>0\) so that any \(n\)-dimensional riemannian manifold \((M,g)\) satisfying \(\textrm{Ric}_g \textrm{diam}(M,g)^2 \geq -\varepsilon(n)\) has first Betti number smaller than or equal to \(n\). In the equality case, \(\textrm{b}_1(M)=n\), Cheeger and Colding showed that then \(M\) has to be bi-Holder homeomorphic to a flat torus. This part can be seen as a stability statement to the rigidity result proven by Bochner, namely, closed riemannian manifolds with nonnegative Ricci curvature and first Betti number equal to their dimension have to be a torus. + The proofs of Gromov and, Cheeger and Colding rely on finding an appropriate subgroup of the abelianized fundamental group to pass to a nice covering space of \(M\) and then study the geometry of the covering. In this talk we will generalize these results to the case of \(RCD(K,N)\) spaces, which is the synthetic notion of riemannian manifolds satisfying \(\text{Ric} \geq K\) and \(\text{dim} \leq N\). This class of spaces include Ricci limit spaces and Alexandrov spaces. + start: + end: + speaker: Raquel Perales (Universidad Nacional Autónoma de México, México), joint with Ilaria Mondello (Université de Paris Est Créteil) and Andrea Mondino (Oxford University) + - titulo: Left-invariant metrics on six-dimensional nilpotent Lie groups + abstract: | + In this talk we determine the moduli space, up to isometric automorphism, of left-invariant metrics on a family of \(6\)-dimensional Lie group \(H\). We also investigate which of these metrics are Hermitian and classify the corresponding complex structures. This talk is based on a joint work with Silvio Reggiani, based on ``The moduli space of left-invariant metrics of a class of six-dimensional nilpotent Lie groups'', arXiv:2011.02854 + start: + end: + speaker: Francisco Vittone (Universidad Nacional de Rosario, Argentina), joint with Silvio Reggiani (Universidad Nacional de Rosario) + - titulo: On the structure of homogeneous Riemannian manifolds with nullity + abstract: | + We will speak about a joint project, with {\it Antonio J. Di Scala} and {\it Francisco Vittone}, for the study of the structure of irreducible homogeneous Riemannian manifolds \(M^n= G/H\) whose curvature tensor has a non-trivial nullity. In a recent paper we developed a general theory to deal with such spaces. By making use of this theory we were able to construct the first non-trivial examples in any dimension. The key fact is the existence of a non-trivial transvection \(X\) at \(p\) (i.e.\((\nabla X)_p = 0\)) such that \(X_p\notin \nu_p\) (the nullity subspace at \(p\)), but the Jacobi operator of \(X_p\) is zero. The nullity distribution \(\nu\) is highly non-homogeneus in the sense that no non-trivial Killing field lie in \(\nu\) (an so \(\nu\) is not given by the orbits of an isometry subgroup of \(G\)). One has that the Lie algebra \(\mathfrak g\) of \(G\) is never reductive. The co-nullity index \(k\) must be always at least \(3\) and if \(k=3\), then \(G\) must be solvable and \(H\) trivial. The leaves of the nullity foliation \(\mathcal F\) are closed and isometric to a Euclidean space. Moreover, the stabilizer of a given leaf acts effectively on the quotient space \(M/\mathcal F\) and so it does not admit a Riemannian \(G\)-invariant metric. Our main new result is that the so-called {\it adapted} tranvections lie in an abelian ideal \(\mathfrak a\) of \(\mathfrak g\). The distribution \(q\mapsto \mathfrak a .q + \nu _q\) is integrable and does not depend on the presentation group \(G\) and so it defines a geometric foliation \(\hat {\mathcal F}\) on \(M\) (by taking the clausure of the leaves). Moreover, the nullity \(\nu\) is parallel along any leaf of \(\hat {\mathcal F}\) and the projection to the quotient space \(M/ \hat {\mathcal F}\) is a Riemannian submersion. We intend to relate the geometry of \(M\) to that of this quotient. + start: + end: + speaker: Carlos E. Olmos (Universidad Nacional de Córdoba, Argentina) + - titulo: TBA + abstract: | + TBA + start: + end: + speaker: Luiz San Martin (Universidade Estadual de Campinas, Brasil) + - titulo: TBA + abstract: | + TBA + start: + end: + speaker: Jorge Lauret (Universidad Nacional de Córdoba, Argentina) - sesion: Comunicación de las matemáticas en América Latina organizadores: - nombre: Javier Elizondo mail: javier@im.unam.mx - nombre: Eduardo Sáenz de Cabezón mail: eduardo.saenz-de-cabezon@unirioja.es + charlas: + - titulo: Círculos Matemáticos, por el fomento al razonamiento propio... + abstract: | + Desde los inicios de nuestra educación, la básica y en el hogar, se tiene una marcada tendencia a sancionar el error; además, producto de nuestros tiempos, se tiene la creencia de que la velocidad es un atributo per se. Esta combinación, en la que se sanciona el error y se induce a aprender las cosas a velocidad, lleva a los infantes a recurrir a la memorización indiscriminada; ello paulatinamente alimenta la inseguridad, la fragilidad y a la larga, la inmovilidad. + Es importante entender que cada individuo requiere de tiempos distintos para hacer suya una idea y desarrollarla, y entender también que errar es parte sustancial del aprendizaje. Comprender las cosas a profundidad no sólo favorece la seguridad del ser humano, sino que fomenta la existencia y desarrollo de perspectivas distintas. + Ante esta situación, el objetivo principal del proyecto de Círculos Matemáticos del Instituto de Matemáticas es despertar e infundir en los jóvenes la confianza en su propio razonamiento, así como el respeto de sus propios tiempos. + Desplazar la noción de éxito basada en la velocidad y las actitudes competitivas, y comprender que el error es algo inherente al proceso de aprendizaje, hace que los estudiantes recuperen la confianza en sí mismos y se atrevan a proponer y explorar caminos. Esto concierne no sólo a las matemáticas sino a la vida misma. + start: + end: + speaker: Laura Ortíz (Universidad Nacional de México, México) + - titulo: El abrazo del escutoide + abstract: | + Sin duda una de las razones más poderosas para insistir en la comunicación de las matemáticas la público general es, al menos para mí, la urgente necesidad de popularización de estas para, sobre todo, acabar con los prejuicios que, tradicionalmente, se han tenido sobre ellas y que conducen, inexorablemente, a la ansiedad de los estudiantes desde las primeras etapas de la educación. Pero, dejando a un lado esta labor del investigador para la sociedad, la divulgación de las matemáticas también pone el trabajo de los investigadores en un escaparate para investigadores de otros ámbitos y puede conducir a la colaboración interdisciplinar. En esta charla les hablaré de cómo la divulgación de la geometría me llevó, nos llevó, a descubrir una nueva forma geométrica mirando a las glándulas salivares de la mosca de la fruta. + start: + end: + speaker: Clara I. Grima (Universidad de Sevilla, España) + - titulo: 'Festival de Matemáticas: la experiencia chilena' + abstract: | + El Festival de Matemáticas de Chile nació a fines del 2016 como un evento satélite al "Primer Encuentro Conjunto de Matemáticas en Chile y Argentina". Es administrado directamente por la Sociedad de Matemática de Chile y se ha presentado en nueve localidades diferentes del país. En 2020 y 2021 tuvo dos versiones en línea debido a la pandemia COVID19. Se entregará un reporte de la organización de estas actividades, del proyecto de elaboración de material didáctico para las escuelas, y de los planes a futuro de la actividad. + start: + end: + speaker: Andrés Navas (Universidad de Santiago de Chile, Chile) + - titulo: La Comisión de Visibilidad de la Unión Matemática Argentina + abstract: | + A fines del 2015 se creo la Comisión de Visibilidad de la Unión Matemática Argentina, conmigo a cargo hasta fines del 2019. En esta charla les contare lo que esta comisión intento/intenta realizar y mi lectura de los resultados obtenidos. Se tratará más de un intercambio de ideas que de una charla en sí. + start: + end: + speaker: Teresa Krick (Universidad de Buenos Aires, Argentina) + - titulo: La divulgación matemática como un instrumento de integración social + abstract: | + Todas las sociedades están formadas por tradiciones, cultura y sobre todo fiestas que crean comunidad y más aún fortalecen el tejido social. De muchas maneras, pareciera que las matemáticas han quedado al margen de estas tradiciones y de la cultura de la sociedades, incluso en muchos casos se han estigmatizado, generando un rechazo per se. + En esta charla platicaré de la importancia de integrar las matemáticas en las tradiciones, en la cultura, y de algunas estrategias que se han desarrollado en el estado de Oaxaca con este fin, a través de las Olimpiadas de matemáticas y el Programa Oaxaqueño de Fortalecimiento a la Educación (PROFE). + start: + end: + speaker: Bruno Aaron Cisneros de la Cruz (Universidad Nacional Autónoma de México, México) + - titulo: Matemáticas (no solo) en las redes + abstract: | + En los últimos tiempos hemos sido testigos de nuevas formas de divulgación de las matemáticas. Nuevas en cuanto a los formatos y (tal vez) en cuanto a los públicos. En esta charla hablaré sobre una experiencia particular en este ámbito, tanto en España como en Latinoamérica. + start: + end: + speaker: Eduardo Sáenz De Cabezón Irigaray (Universidad de la Rioja, España) + - titulo: Haciendo comunidad en tiempos de confinamiento + abstract: | + En medio de la incertidumbre de la pandemia de COVID-19 y en el boom de la virtualidad, se han abierto puertas de comunicación entre divulgadores de matemáticas hispanoparlantes en América Latina y en España. En este camino se han organizado encuentros directamente centrados en crear puentes sólidos de colaboración entre distintos grupos. + Durante esta conferencia, presentaré el trabajo que hemos hecho en este sentido Ágata Timón (Instituto de Ciencias Matemáticas), Andrés Navas (Universidad de Chile), Aubin Arroyo (UNAM), Beatriz Vargas (UNAM) y yo. También me gustaría plantear y oír propuestas de caminos para fortalecer la colaboración entre quienes trabajamos divulgando las matemáticas en español. + start: + end: + speaker: Lucía López de Medrano (Universidad Nacional Autónoma de México, México) + - titulo: Narrativa de no-ficción con temas matemáticos + abstract: | + En esta charla trataré de abrir la discusión sobre qué significa la narrativa de no-ficción dentro de la divulgación de las matemáticas. + En este tipo de narrativa se desarrolla una historia en la que se incluye contenido matemático. Surgen varias preguntas: ¿la historia tiene más importancia que el contenido matemático?, ¿se debe de ser muy riguroso con el contenido matemático? + Estas preguntas están relacionadas con la postura de Humberto Eco, quien en un Postcript a su novela "El nombre de la rosa" criticó fuertemente lo que se conoce como Salgarismo: suspender una narración dentro de la trama para realizar una (extensa) explicación científica. + No se busca dar una respuesta a estas preguntas sino plantear algunas posturas y preguntas que permitan avanzar en una discusión sobre este tema. + start: + end: + speaker: Javier Elizondo (Universidad Nacional Autónoma de México, México) + - titulo: La divulgación de las matemáticas en Costa Rica + abstract: | + Se compartirá la experiencia generada por un grupo de personas que nos hemos dedicado a la divulgación de las matemáticas, con el apoyo de organizaciones no gubernamentales, empresa privada y universidades estatales. En este esfuerzo se han realizado diversas actividades que han tenido un impacto sostenido e interesante. En primer lugar, la producción de “Matemática por un minuto”, una serie de programas para la radio con duración de un minuto cada uno que se transmitieron a nivel nacional dos veces al día y su evolución inesperada hacia videos y posteriormente hacia un libro titulado “Las matemáticas de lo cotidiano”, en sus versiones en español e inglés, así como su edición física y e-book. + Además, el diseño, construcción, experiencias, evolución y clonación del “Museo viajante de ciencias y matemática (MUCYM)” y su impacto en la divulgación de las matemáticas a lo largo y ancho de nuestro país, sobre todo a sectores menos favorecidos de su población. + Por último, la organización de congresos académicos, como el evento bianual “Festival Internacional de Matemáticas” en que se inició en 1998 En las doce ediciones que se han realizado, se amalgamaron mágicamente propuestas didácticas, talleres, ponencias y conferencias con juegos, poesía, museos y teatro entre otros que motivan una participación de profesores, estudiantes y varios grupos sociales que propician el aprendizaje activo de esta bella ciencia, las matemáticas. + start: + end: + speaker: Manuel Murillo Tsijli (Instituto Tecnológico de Costa Rica, Costa Rica)