From e91bb22f9b138d883019e307c19fd59925f50f7c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Germ=C3=A1n=20Correa?= Date: Fri, 3 Sep 2021 15:01:24 -0300 Subject: [PATCH] agrega

en abstracts completa hasta sesion 27 inclusive --- data/sesiones.yml | 296 ++++++++++++++++++++--------------------- public/style/style.css | 2 +- 2 files changed, 149 insertions(+), 149 deletions(-) diff --git a/data/sesiones.yml b/data/sesiones.yml index 1f276bd..577345e 100644 --- a/data/sesiones.yml +++ b/data/sesiones.yml @@ -130,40 +130,40 @@ mail: M.Hyland@dpmms.cam.ac.uk charlas: - titulo: Random! - abstract: Everyone has an intuitive idea about what is randomness, often associated with ``gambling'' or ``luck''. Is there a mathematical definition of randomness? Are there degrees of randomness? Can we give examples of randomness? Can a computer produce a sequence that is truly random? What is the relation between randomness and logic? In this talk I will talk about these questions and their answers. + abstract:

Everyone has an intuitive idea about what is randomness, often associated with ``gambling'' or ``luck''. Is there a mathematical definition of randomness? Are there degrees of randomness? Can we give examples of randomness? Can a computer produce a sequence that is truly random? What is the relation between randomness and logic? In this talk I will talk about these questions and their answers.

start: 2021-09-14T16:45-0300 end: 2021-09-14T17:30-0300 speaker: Verónica Becher (Universidad de Buenos Aires, Argentina) - titulo: Relating logical approaches to concurrent computation abstract: | - This talk will present ongoing work towards the description and study of concurrent interaction in proof theory. - Type systems that are designed to ensure behavioural properties of concurrent processes (input/output regimes, lock-freeness) generally have unclear logical meanings. Conversely, proofs-as-programs correspondences for processes (e.g. with session types) tend to impose very functional behaviour and little actual concurrency. Besides, relationships between type systems and denotational models of concurrency are rarely established. - A possible reason for this state of things is the ambiguous status of non-determinism in logic and the importance of scheduling concerns in models of concurrency, to which traditional proof theory is not accustomed. Unifying logical approaches in a consistent framework requires to put a focus on these issues, and this talk will propose, building on recent developments in proof theory, in the veins of linear logic and classical realizability. +

This talk will present ongoing work towards the description and study of concurrent interaction in proof theory.

+

Type systems that are designed to ensure behavioural properties of concurrent processes (input/output regimes, lock-freeness) generally have unclear logical meanings. Conversely, proofs-as-programs correspondences for processes (e.g. with session types) tend to impose very functional behaviour and little actual concurrency. Besides, relationships between type systems and denotational models of concurrency are rarely established.

+

A possible reason for this state of things is the ambiguous status of non-determinism in logic and the importance of scheduling concerns in models of concurrency, to which traditional proof theory is not accustomed. Unifying logical approaches in a consistent framework requires to put a focus on these issues, and this talk will propose, building on recent developments in proof theory, in the veins of linear logic and classical realizability.

start: 2021-09-13T15:00-0300 end: 2021-09-13T15:45-0300 speaker: Emmanuel Beffara (Université Grenoble Alpes, Francia) - titulo: A framework to express the axioms of mathematics - abstract: 'The development of computer-checked formal proofs is a major step forward in the endless quest for mathematical rigor. But it also has a negative aspect: the multiplicity of systems brought a multiplicity of theories in which these formal proofs are expressed. We propose to define these theories in a common logical framework, called Dedukti. Some axioms are common to the various theories and some others are specific, just like some axioms are common to all geometries and some others are specific. This logical framework extends predicate logic in several ways and we shall discuss why predicate logic must be extended to enable the expression of these theories.' + abstract: '

The development of computer-checked formal proofs is a major step forward in the endless quest for mathematical rigor. But it also has a negative aspect: the multiplicity of systems brought a multiplicity of theories in which these formal proofs are expressed. We propose to define these theories in a common logical framework, called Dedukti. Some axioms are common to the various theories and some others are specific, just like some axioms are common to all geometries and some others are specific. This logical framework extends predicate logic in several ways and we shall discuss why predicate logic must be extended to enable the expression of these theories.

' start: 2021-09-14T15:45-0300 end: 2021-09-14T16:30-0300 speaker: Gilles Dowek (Institut de la Recherche en Informatique et Automatique, Francia) - titulo: Generalized Algebraic Theories and Categories with Families - abstract: We give a new syntax independent definition of the notion of a finitely presented generalized algebraic theory as an initial object in a category of categories with families (cwfs) with extra structure. To this end we define inductively how to build a valid signature \(\Sigma\) for a generalized algebraic theory and the associated category \(\textrm{CwF}_{\Sigma}\) of cwfs with a \(\Sigma\)-structure and cwf-morphisms that preserve \(\Sigma\)-structure on the nose. Our definition refers to the purely semantic notions of uniform family of contexts, types, and terms. Furthermore, we show how to syntactically construct initial cwfs with \(\Sigma\)-structures. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of dependent type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual categories with families. + abstract:

We give a new syntax independent definition of the notion of a finitely presented generalized algebraic theory as an initial object in a category of categories with families (cwfs) with extra structure. To this end we define inductively how to build a valid signature \(\Sigma\) for a generalized algebraic theory and the associated category \(\textrm{CwF}_{\Sigma}\) of cwfs with a \(\Sigma\)-structure and cwf-morphisms that preserve \(\Sigma\)-structure on the nose. Our definition refers to the purely semantic notions of uniform family of contexts, types, and terms. Furthermore, we show how to syntactically construct initial cwfs with \(\Sigma\)-structures. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of dependent type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual categories with families.

start: 2021-09-13T15:45-0300 end: 2021-09-13T16:30-0300 speaker: Peter Dybjer (Chalmers University of Technology, Suecia), joint with Marc Bezem, Thierry Coquand, and Martin Escardo - titulo: On the instability of the consistency operator - abstract: We examine recursive monotonic functions on the Lindenbaum algebra of EA. We prove that no such function sends every consistent \(\varphi\), to a sentence with deductive strength strictly between \(\varphi\) and \(\textit{Con}(\varphi)\). We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function \(f\), if there is an iterate of \(\textit{Con}\) that bounds \(f\) everywhere, then \(f\) must be somewhere equal to an iterate of \(\textit{Con}\). + abstract:

We examine recursive monotonic functions on the Lindenbaum algebra of EA. We prove that no such function sends every consistent \(\varphi\), to a sentence with deductive strength strictly between \(\varphi\) and \(\textit{Con}(\varphi)\). We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function \(f\), if there is an iterate of \(\textit{Con}\) that bounds \(f\) everywhere, then \(f\) must be somewhere equal to an iterate of \(\textit{Con}\).

start: 2021-09-13T17:30-0300 end: 2021-09-13T18:15-0300 speaker: Antonio Montalbán (Berkeley University of California, Estados Unidos), joint work with James Walsh - titulo: Readers by name, presheaves by value - abstract: Presheaves are an ubiquitary model construction used everywhere in logic, particularly in topos theory. It is therefore tempting to port them to the similar but slightly different context of type theory. Unfortunately, it turns out that there are subtle issues with the built-in computation rules of the latter, which we will expose. As an alternative, we will describe a new structure that is much better behaved in an intensional setting, but categorically equivalent to presheaves in an extensional one. Such a structure is motivated by considerations stemming from the study of generic side-effects in programming language theory, shedding a new light on the fundamental nature of such a well-known object. + abstract:

Presheaves are an ubiquitary model construction used everywhere in logic, particularly in topos theory. It is therefore tempting to port them to the similar but slightly different context of type theory. Unfortunately, it turns out that there are subtle issues with the built-in computation rules of the latter, which we will expose. As an alternative, we will describe a new structure that is much better behaved in an intensional setting, but categorically equivalent to presheaves in an extensional one. Such a structure is motivated by considerations stemming from the study of generic side-effects in programming language theory, shedding a new light on the fundamental nature of such a well-known object.

start: 2021-09-14T15:00-0300 end: 2021-09-14T15:45-0300 speaker: Pierre-Marie Pédrot (Institut de la Recherche en Informatique et Automatique, Francia) - titulo: Reversible computation and quantum control - abstract: One perspective on quantum algorithms is that they are classical algorithms having access to a special kind of memory with exotic properties. This perspective suggests that, even in the case of quantum algorithms, the control flow notions of sequencing, conditionals, loops, and recursion are entirely classical. There is however, another notion of control flow, that is itself quantum. This purely quantum control flow is however not well-understood. In this talk, I will discuss how to retrieve some understanding of it with a detour through reversible computation. This will allow us to draw links with the logic \(\mu\)MALL, pointing towards a Curry-Howard isomorphism. + abstract:

One perspective on quantum algorithms is that they are classical algorithms having access to a special kind of memory with exotic properties. This perspective suggests that, even in the case of quantum algorithms, the control flow notions of sequencing, conditionals, loops, and recursion are entirely classical. There is however, another notion of control flow, that is itself quantum. This purely quantum control flow is however not well-understood. In this talk, I will discuss how to retrieve some understanding of it with a detour through reversible computation. This will allow us to draw links with the logic \(\mu\)MALL, pointing towards a Curry-Howard isomorphism.

start: 2021-09-13T16:45-0300 end: 2021-09-13T17:30-0300 speaker: Benoît Valiron (CentraleSupélec, Francia) @@ -229,69 +229,69 @@ mail: agnelli@famaf.unc.edu.ar charlas: - titulo: Synergistic multi-spectral CT reconstruction with directional total variation - abstract: This work considers synergistic multi-spectral CT reconstruction where information from all available energy channels is combined to improve the reconstruction of each individual channel. We propose to fuse these available data (represented by a single sinogram) to obtain a polyenergetic image that keeps structural information shared by the energy channels with an increased signal-to-noise ratio. This new image is used as prior information during a channel-by-channel minimization process through the directional total variation. We analyze the use of directional total variation within variational regularization and iterative regularization. Our numerical results on simulated and experimental data show improvements in terms of image quality and in computational speed. + abstract:

This work considers synergistic multi-spectral CT reconstruction where information from all available energy channels is combined to improve the reconstruction of each individual channel. We propose to fuse these available data (represented by a single sinogram) to obtain a polyenergetic image that keeps structural information shared by the energy channels with an increased signal-to-noise ratio. This new image is used as prior information during a channel-by-channel minimization process through the directional total variation. We analyze the use of directional total variation within variational regularization and iterative regularization. Our numerical results on simulated and experimental data show improvements in terms of image quality and in computational speed.

start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Evelyn Cueva (Escuela Politécnica Nacional, Ecuador) - titulo: Bayesian Approach Helmholtz Inverse Problem - abstract: This work present an implementation for solving the Helmholtz inverse problem via Finite Element Methods and Bayesian inference theory. The solution of the inverse problem is a posterior distribution based on a number observation data sets, the forward problem governing the state solution, and a statistical prior distribution which describes the behavior of the uncertainty linked to the parameters. + abstract:

This work present an implementation for solving the Helmholtz inverse problem via Finite Element Methods and Bayesian inference theory. The solution of the inverse problem is a posterior distribution based on a number observation data sets, the forward problem governing the state solution, and a statistical prior distribution which describes the behavior of the uncertainty linked to the parameters.

start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Lilí Guadarrama (Centro de Investigación en Matemáticas, México) - titulo: Partial data Photoacoustic Tomography in unbounded domains abstract: | - Photoacoustic Tomography is a promising hybrid medical imaging modality that is able to generate high-resolution and high-contrast images, by exploiting the coupling of electromagnetic pulses (in the visible region) and ultrasound waves via de photoacoustic effect. - In this talk I will introduce the modality and focus on the ultrasound propagation part which is mathematically modeled as an inverse initial source problem for the wave equation. We will pose this problem in a setting consisting of an unbounded domain with boundary, with observation taking place in a subset of it. We will address the question of uniqueness, stability and reconstruction of the initial source of acoustic waves. +

Photoacoustic Tomography is a promising hybrid medical imaging modality that is able to generate high-resolution and high-contrast images, by exploiting the coupling of electromagnetic pulses (in the visible region) and ultrasound waves via de photoacoustic effect.

+

In this talk I will introduce the modality and focus on the ultrasound propagation part which is mathematically modeled as an inverse initial source problem for the wave equation. We will pose this problem in a setting consisting of an unbounded domain with boundary, with observation taking place in a subset of it. We will address the question of uniqueness, stability and reconstruction of the initial source of acoustic waves.

start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Benjamín Palacios (Pontificia Universidad Católica de Chile, Chile) - titulo: Particle Source Estimation via Analytical Formulations of the Adjoint Flux - abstract: Numerous applications of research interest may benefit from the study and development of particle sources estimation methods, such as non-destructive material identification and nuclear safety. In such problems, we are concerned with estimating the spatial distribution and intensity of internal sources of neutral particles. In this context, we use the adjoint to the transport operator to derive a linear model that relates the absorption rate measurements of a series of internal particle detectors with the coefficients of the source expansion on a given basis. We consider both one-dimensional energy-dependent and two-dimensional (\(XY\)-geometry) monoenergetic transport problems. We use the adjoint version of the Analytical Discrete Ordinates (ADO) method to write spatially explicit solutions for the adjoint flux for one-dimensional problems. We consider a nodal version of the adjoint ADO method to writing similar explicit expressions for the \(x\) and \(y\) averaged adjoint fluxes for the two-dimensional case. In both cases, we use explicit expressions to derive closed-form formulas to the absorption rate. Then, we apply the iterated Tikhonov algorithm to estimate the sources from noisy measurements and a Bayesian approach through the Metropolis-Hastings algorithm. We successfully estimated one-dimensional polynomial and piecewise constant localized energy dependent sources and two-dimensional piecewise constant localized sources from noise measurements. In either case, the use of closed-form expressions allowed a relevant reduction in computational time. + abstract:

Numerous applications of research interest may benefit from the study and development of particle sources estimation methods, such as non-destructive material identification and nuclear safety. In such problems, we are concerned with estimating the spatial distribution and intensity of internal sources of neutral particles. In this context, we use the adjoint to the transport operator to derive a linear model that relates the absorption rate measurements of a series of internal particle detectors with the coefficients of the source expansion on a given basis. We consider both one-dimensional energy-dependent and two-dimensional (\(XY\)-geometry) monoenergetic transport problems. We use the adjoint version of the Analytical Discrete Ordinates (ADO) method to write spatially explicit solutions for the adjoint flux for one-dimensional problems. We consider a nodal version of the adjoint ADO method to writing similar explicit expressions for the \(x\) and \(y\) averaged adjoint fluxes for the two-dimensional case. In both cases, we use explicit expressions to derive closed-form formulas to the absorption rate. Then, we apply the iterated Tikhonov algorithm to estimate the sources from noisy measurements and a Bayesian approach through the Metropolis-Hastings algorithm. We successfully estimated one-dimensional polynomial and piecewise constant localized energy dependent sources and two-dimensional piecewise constant localized sources from noise measurements. In either case, the use of closed-form expressions allowed a relevant reduction in computational time.

start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Liliane Basso Barichello (Universidade Federal do Rio Grande do Sul, Brasil) - titulo: Inverse scattering for the Jacobi system using input data containing transmission eigenvalues abstract: | - The Jacobi system with the Dirichlet boundary condition is considered on a half-line lattice with real-valued coefficients including weight factors. The inverse problem of recovery of the coefficients is analyzed by using certain input data sets containing the transmission eigenvalues. - This is joint work with T. Aktosun of University of Texas at Arlington and V. G. Papanicolaou of National Technical University of Athens. +

The Jacobi system with the Dirichlet boundary condition is considered on a half-line lattice with real-valued coefficients including weight factors. The inverse problem of recovery of the coefficients is analyzed by using certain input data sets containing the transmission eigenvalues.

+

This is joint work with T. Aktosun of University of Texas at Arlington and V. G. Papanicolaou of National Technical University of Athens.

start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Abdon Choque Rivero (Universidad Michoacana de San Nicolás de Hidalgo, México) - titulo: Identificación del coeficiente de velocidad local en un modelo bi-direccional para la propagación de ondas en la superficie de un canal con fondo variable abstract: | - Un problema inverso es aquel donde se requiere determinar el valor de algunos parámetros de un modelo a partir de datos observados. El estudio de este tipo de problemas es de mucha importancia y aparece en diversas ramas de la ciencia y de las Matemáticas. En esta charla consideramos el problema inverso de identificación del coeficiente de velocidad lineal local en un sistema de tipo Boussinesq linealizado de tipo dispersivo deducido por Quintero y Muñoz (2004), a partir de las mediciones \(V_T(x), N_T(x)\) en un tiempo final \(T\), de la velocidad de las partículas en una profundidad fija y la amplitud de la onda, que se propaga en la superficie de un canal raso con fondo variable. - El problema inverso es reformulado como un problema de optimización con restricciones, donde la funcional objetivo es de tipo usado en métodos de mínimos cuadrados junto con una regularización del tipo introducido por Tikhonov (1963). El proceso de minimización del funcional Tikhonov adoptado se realiza utilizando el método iterativo quasi-Newton L-BFGS-B, introducido por Byrd et al. (1995). Se presentan simulaciones numéricas para mostrar la convergencia y robustez del método numérico propuesto, a\'un en el caso en que el coeficiente posee discontinuidades, o existe ruido en la medici\'on de los datos de entrada \(V_T(x), N_T(x)\). - Investigación realizada con el apoyo de la Universidad del Valle mediante proyecto C.I. 71235. - Referencias: - Tikhonov, A.N. (1963): Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl. 4, 1035-1038. - Byrd, R.H., Lu, P., Nocedal, J. (1995): A limited memory algorithm for Bound Constrained Optimization. SIAM J. Sci. Stat. Comp. 16 (5): 1190-1208. - Quintero, J.R. , Muñoz, J.C. (2004): Existence and uniqueness for a Boussinesq system with a disordered forcing. Methods Appl. Anal., 11 (1): 015-032. +

Un problema inverso es aquel donde se requiere determinar el valor de algunos parámetros de un modelo a partir de datos observados. El estudio de este tipo de problemas es de mucha importancia y aparece en diversas ramas de la ciencia y de las Matemáticas. En esta charla consideramos el problema inverso de identificación del coeficiente de velocidad lineal local en un sistema de tipo Boussinesq linealizado de tipo dispersivo deducido por Quintero y Muñoz (2004), a partir de las mediciones \(V_T(x), N_T(x)\) en un tiempo final \(T\), de la velocidad de las partículas en una profundidad fija y la amplitud de la onda, que se propaga en la superficie de un canal raso con fondo variable.

+

El problema inverso es reformulado como un problema de optimización con restricciones, donde la funcional objetivo es de tipo usado en métodos de mínimos cuadrados junto con una regularización del tipo introducido por Tikhonov (1963). El proceso de minimización del funcional Tikhonov adoptado se realiza utilizando el método iterativo quasi-Newton L-BFGS-B, introducido por Byrd et al. (1995). Se presentan simulaciones numéricas para mostrar la convergencia y robustez del método numérico propuesto, a\'un en el caso en que el coeficiente posee discontinuidades, o existe ruido en la medici\'on de los datos de entrada \(V_T(x), N_T(x)\).

+

Investigación realizada con el apoyo de la Universidad del Valle mediante proyecto C.I. 71235.

+

Referencias:
+ Tikhonov, A.N. (1963): Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl. 4, 1035-1038.
+ Byrd, R.H., Lu, P., Nocedal, J. (1995): A limited memory algorithm for Bound Constrained Optimization. SIAM J. Sci. Stat. Comp. 16 (5): 1190-1208.
+ Quintero, J.R. , Muñoz, J.C. (2004): Existence and uniqueness for a Boussinesq system with a disordered forcing. Methods Appl. Anal., 11 (1): 015-032.

start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Juan Carlos Muñoz Grajales (Universidad del Valle, Colombia) - titulo: Joint curvature-driven diffusion and weighted anisotropic total variation regularization for local inpainting - abstract: An image inpainting problem consists of restoring an image from a (possibly noisy) observation, in which data from one or more regions is missing. Several inpainting models to perform this task have been developed, and although some of them perform reasonably well in certain types of images, quite a few issues are yet to be sorted out. For instance, if the original image is expected to be smooth, inpainting can be performed with reasonably good results by modeling the solution as the result of a diffusion process using the heat equation. For non-smooth images, however, such an approach is far from being satisfactory. On the other hand, Total Variation (TV) inpainting models based on high order PDE diffusion equations can be used whenever edge restoration is a priority. More recently, the introduction of spatially variant conductivity coefficients on these models, such as in the case of Curvature-Driven Diffusions (CDD), has allowed inpainted images with well defined edges and enhanced object connectivity. The CDD approach, nonetheless, is not suitable wherever the image is smooth, as it tends to produce piecewise constant solutions. Based upon this, we propose using CDD to gather a-priori information used at a second step in a weighted anisotropic mixed Tikhonov plus total- variation model that allows for both edge preservation and object connectivity while precluding the staircasing effect that all pure TV-based methods entail. Comparisons between the results of the implemented models will be illustrated by several computed examples, along with performance measures. + abstract:

An image inpainting problem consists of restoring an image from a (possibly noisy) observation, in which data from one or more regions is missing. Several inpainting models to perform this task have been developed, and although some of them perform reasonably well in certain types of images, quite a few issues are yet to be sorted out. For instance, if the original image is expected to be smooth, inpainting can be performed with reasonably good results by modeling the solution as the result of a diffusion process using the heat equation. For non-smooth images, however, such an approach is far from being satisfactory. On the other hand, Total Variation (TV) inpainting models based on high order PDE diffusion equations can be used whenever edge restoration is a priority. More recently, the introduction of spatially variant conductivity coefficients on these models, such as in the case of Curvature-Driven Diffusions (CDD), has allowed inpainted images with well defined edges and enhanced object connectivity. The CDD approach, nonetheless, is not suitable wherever the image is smooth, as it tends to produce piecewise constant solutions. Based upon this, we propose using CDD to gather a-priori information used at a second step in a weighted anisotropic mixed Tikhonov plus total- variation model that allows for both edge preservation and object connectivity while precluding the staircasing effect that all pure TV-based methods entail. Comparisons between the results of the implemented models will be illustrated by several computed examples, along with performance measures.

start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Ruben D. Spies (Instituto de Matemática Aplicada del Litoral, Argentina) - titulo: A Splitting Strategy for the Calibration of Jump-Diffusion Models - abstract: We present a splitting strategy to identify simultaneously the local-volatility surface and the jump-size distribution from quoted European prices. The underlying is a jump-diffusion driven asset with time and price dependent volatility. Our approach uses a forward partial-integro-differential equation for the option prices to produce a parameter-to-solution map. The corresponding inverse problem is then solved by Tikhonov-type regularization combined with a splitting strategy. + abstract:

We present a splitting strategy to identify simultaneously the local-volatility surface and the jump-size distribution from quoted European prices. The underlying is a jump-diffusion driven asset with time and price dependent volatility. Our approach uses a forward partial-integro-differential equation for the option prices to produce a parameter-to-solution map. The corresponding inverse problem is then solved by Tikhonov-type regularization combined with a splitting strategy.

start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Vinicus V. Albani (Universidade Federal de Santa Catarina, Brasil) - titulo: Lipschitz Stability for the Backward Heat Equation and its application to Lightsheet Fluorescence Microscopy abstract: | - In this talk we will present a result about Lipschitz stability for the reconstruction of a compactly supported initial temperature for the heat equation in \(\mathbb{R}^n\), from measurements along a positive time interval and over an open set containing its support. We apply these results to deduce a Lipschitz stability inequality for the backwards heat equation problem in \(\mathbb{R}\) with measurements on a curve contained in \( \mathbb{R}\times[0,T]\), leading to stability estimates for an inverse problem arising in 2D Fluorescence Microscopy. - This is joint work with Pablo Arratia, Evelyn Cueva, Axel Osses and Benjamin Palacios. +

In this talk we will present a result about Lipschitz stability for the reconstruction of a compactly supported initial temperature for the heat equation in \(\mathbb{R}^n\), from measurements along a positive time interval and over an open set containing its support. We apply these results to deduce a Lipschitz stability inequality for the backwards heat equation problem in \(\mathbb{R}\) with measurements on a curve contained in \( \mathbb{R}\times[0,T]\), leading to stability estimates for an inverse problem arising in 2D Fluorescence Microscopy.

+

This is joint work with Pablo Arratia, Evelyn Cueva, Axel Osses and Benjamin Palacios.

start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Matias Courdurier (Pontificia Universidad Católica, Chile) - titulo: Anatomical atlas of the upper part of the human head for electroencephalography and bioimpedance applications abstract: | - Electrophysiology is the branch of physiology that investigates the electrical properties of biological tissues. Volume conductor problems in cerebral electrophysiology and bioimpedance do not have analytical solutions for nontrivial geometries and require a 3D model of the head and its electrical properties for solving the associated PDEs numerically. - Ideally, the model should be made with patient-specific information. In clinical practice, this is not always the case and an average head model is often used. Also, the electrical properties of the tissues might not be completely known due to natural variability. - A 4D (3D+T) statistical anatomical atlas of the electrical properties of the upper part of the human head for cerebral electrophysiology and bioimpedance applications is presented. The atlas is constructed based on MRI images of human individuals and comprises the electrical properties of the main internal structures and can be adjusted for specific electrical frequencies. The atlas also comprises a time-varying model of arterial brain circulation, based on the solution of the Navier-Stokes equations. - The atlas is an important tool for in silico studies on cerebral circulation and electrophysiology that require statistically consistent data, e.g., machine learning, sensitivity analyses, and as a benchmark to test inverse problem solvers. The atlas can also be used as statistical prior information for inverse problems in electrophysiology. +

Electrophysiology is the branch of physiology that investigates the electrical properties of biological tissues. Volume conductor problems in cerebral electrophysiology and bioimpedance do not have analytical solutions for nontrivial geometries and require a 3D model of the head and its electrical properties for solving the associated PDEs numerically.

+

Ideally, the model should be made with patient-specific information. In clinical practice, this is not always the case and an average head model is often used. Also, the electrical properties of the tissues might not be completely known due to natural variability.

+

A 4D (3D+T) statistical anatomical atlas of the electrical properties of the upper part of the human head for cerebral electrophysiology and bioimpedance applications is presented. The atlas is constructed based on MRI images of human individuals and comprises the electrical properties of the main internal structures and can be adjusted for specific electrical frequencies. The atlas also comprises a time-varying model of arterial brain circulation, based on the solution of the Navier-Stokes equations.

+

The atlas is an important tool for in silico studies on cerebral circulation and electrophysiology that require statistically consistent data, e.g., machine learning, sensitivity analyses, and as a benchmark to test inverse problem solvers. The atlas can also be used as statistical prior information for inverse problems in electrophysiology.

start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Fernando Moura da Silva (Universidade Federal do ABC, Brasil y University of Helsinki, Finlandia) @@ -734,31 +734,31 @@ charlas: - titulo: Conserved quantities and the existence of (twisted) Poisson brackets in nonholonomic mechanics abstract: | - In this talk, I will start by explaining the nonhamiltonian nature of nonholonomic systems, and then we will study the "hamiltonization problem" from a geometric standpoint. By making use of symmetries and suitable first integrals of the system, we will explicitly define a new bracket on the reduced space codifying the nonholonomic dynamics that, in many examples, is a genuine Poisson bracket making the system hamiltonian (in general, the new bracket carries an almost symplectic foliation determined by the first integrals). This is a joint work with Luis Yapu-Quispe. +

In this talk, I will start by explaining the nonhamiltonian nature of nonholonomic systems, and then we will study the "hamiltonization problem" from a geometric standpoint. By making use of symmetries and suitable first integrals of the system, we will explicitly define a new bracket on the reduced space codifying the nonholonomic dynamics that, in many examples, is a genuine Poisson bracket making the system hamiltonian (in general, the new bracket carries an almost symplectic foliation determined by the first integrals). This is a joint work with Luis Yapu-Quispe.

start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Paula Balseiro (Universidad Federal Fluminense, Brasil) - titulo: From retraction maps to geometric integrators for optimal control problems abstract: | - Retraction maps are used in many research fields such as approximation of trajectories of differential equations, optimization theory, interpolation theory etc. In this talk we will review the concept of retraction map in differentiable manifolds to generalize it to obtain an extended retraction map from the tangent bundle of the configuration manifold to two copies of this manifold. After suitably lifting the new retraction map to the cotangent bundle, the typical phase space for Hamiltonian mechanical systems, we will be able to define geometric integrators for optimal control problems. This is a joint work with David Martín de Diego. +

Retraction maps are used in many research fields such as approximation of trajectories of differential equations, optimization theory, interpolation theory etc. In this talk we will review the concept of retraction map in differentiable manifolds to generalize it to obtain an extended retraction map from the tangent bundle of the configuration manifold to two copies of this manifold. After suitably lifting the new retraction map to the cotangent bundle, the typical phase space for Hamiltonian mechanical systems, we will be able to define geometric integrators for optimal control problems. This is a joint work with David Martín de Diego.

start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: María Barbero Liñán (Universidad Politécnica de Madrid, España) - titulo: Relación entre la teoría de campos de Chern-Simons y relatividad general en dimensión \(3\), desde el punto de vista de los problemas variacionales de Griffiths abstract: | - La teoría de campos de Chern-Simons es una teoría de campos topológica, y puede formularse sobre cualquier fibrado principal cuyo grupo de estructura admita una forma bilineal invariante. Consideremos un fibrado principal \(\pi:P\to M\) con grupo de estructura \(G\) y sea \(K\subset G\) un subgrupo; tomemos la familia de problemas variacionales de Chern-Simons sobre todos los subfibrados de \(P\) con fibra \(K\). En la presente charla explicaremos cómo esta familia de problemas variacionales puede codificarse mediante un único principio variacional de tipo Griffiths. Finalmente, utilizaremos el problema variacional construído para entender desde un punto de vista geométrico la correspondencia entre gravedad en dimensión \(2+1\) y la teoría de Chern-Simons. +

La teoría de campos de Chern-Simons es una teoría de campos topológica, y puede formularse sobre cualquier fibrado principal cuyo grupo de estructura admita una forma bilineal invariante. Consideremos un fibrado principal \(\pi:P\to M\) con grupo de estructura \(G\) y sea \(K\subset G\) un subgrupo; tomemos la familia de problemas variacionales de Chern-Simons sobre todos los subfibrados de \(P\) con fibra \(K\). En la presente charla explicaremos cómo esta familia de problemas variacionales puede codificarse mediante un único principio variacional de tipo Griffiths. Finalmente, utilizaremos el problema variacional construído para entender desde un punto de vista geométrico la correspondencia entre gravedad en dimensión \(2+1\) y la teoría de Chern-Simons.

start: 2021-09-16T17:30 end: 2021-09-17T18:15 speaker: Santiago Capriotti (Universidad Nacional del Sur, Argentina) - titulo: Lie group's exponential curves and the Hamilton-Jacobi theory abstract: | - In this talk we present an extended version of the Hamilton-Jacobi equation (HJE), valid for general dynamical systems defined by vector fields (not only by the Hamiltonian ones), and a result which ensures that, if we have a complete solution of the HJE for a given dynamical system, inside a certain subclass of systems, then such a system can be integrated up to quadratures. Then we apply this result to show that the exponential curves \(\exp\left(\eta\,t\right)\) of any Lie group, which are the integral curves of the left invariant vector fields in the group, can be constructed up to quadratures (unless for certain elements \(\eta\) inside its corresponding Lie algebra). This gives rise to an alternative concrete expression of \(\exp\left(\eta\,t\right)\), different to those that appears in the literature for matrix Lie groups. +

In this talk we present an extended version of the Hamilton-Jacobi equation (HJE), valid for general dynamical systems defined by vector fields (not only by the Hamiltonian ones), and a result which ensures that, if we have a complete solution of the HJE for a given dynamical system, inside a certain subclass of systems, then such a system can be integrated up to quadratures. Then we apply this result to show that the exponential curves \(\exp\left(\eta\,t\right)\) of any Lie group, which are the integral curves of the left invariant vector fields in the group, can be constructed up to quadratures (unless for certain elements \(\eta\) inside its corresponding Lie algebra). This gives rise to an alternative concrete expression of \(\exp\left(\eta\,t\right)\), different to those that appears in the literature for matrix Lie groups.

start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Sergio Grillo (Centro Atómico Bariloche, Argentina) - titulo: Null hyperpolygons and quasi-parabolic Higgs bundles abstract: | - Hyperpolygons spaces are a family of hyperkähler manifolds that can be obtained from coadjoint orbits by hyperkähler reduction. Jointly with L. Godinho, we showed that these spaces are isomorphic to certain families of parabolic Higgs bundles, when a suitable condition between the parabolic weights and the spectra of the coadjoint orbits is satisfied. In analogy to this construction, we introduce two moduli spaces: the moduli spaces of quasi-parabolic \(SL(2,\mathbb{C})\)-Higgs bundles over \(\mathbb{C}\mathbb{P}^1\) on one hand and the null hyperpolygon spaces on the other, and establish an isomorphism between them. Finally we describe the fixed loci of natural involutions defined on these spaces and relate them to the moduli space of null hyperpolygons in the Minkowski 3-space. This is based on joint works with Leonor Godinho. +

Hyperpolygons spaces are a family of hyperkähler manifolds that can be obtained from coadjoint orbits by hyperkähler reduction. Jointly with L. Godinho, we showed that these spaces are isomorphic to certain families of parabolic Higgs bundles, when a suitable condition between the parabolic weights and the spectra of the coadjoint orbits is satisfied. In analogy to this construction, we introduce two moduli spaces: the moduli spaces of quasi-parabolic \(SL(2,\mathbb{C})\)-Higgs bundles over \(\mathbb{C}\mathbb{P}^1\) on one hand and the null hyperpolygon spaces on the other, and establish an isomorphism between them. Finally we describe the fixed loci of natural involutions defined on these spaces and relate them to the moduli space of null hyperpolygons in the Minkowski 3-space. This is based on joint works with Leonor Godinho.

start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Alessia Mandini (Universidade Federal Fluminense, Brasil) @@ -775,13 +775,13 @@ speaker: Juan Carlos Marrero (Universidad de La Laguna, España) - titulo: Discrete variational calculus and accelerated methods in optimization abstract: | - Many of the new developments in machine learning are connected with gradient-based optimization methods. Recently, these methods have been studied using a variational perspective. This has opened up the possibility of introducing variational and symplectic integration methods using geometric integrators. In particular, in thistle, we introduce variational integrators which allows us to derive different methods for optimization. However, since the systems are explicitly time-dependent, the preservation of the symplecticity property occurs solely on the fibers. Finally, using discrete Lagrange-d'Alembert principle we produce optimization methods whose behavior is similar to the classical Nesterov method reducing the oscillations of typical momentum methods. Joint work with Cédric M. Campos and Alejandro Mahillo. +

Many of the new developments in machine learning are connected with gradient-based optimization methods. Recently, these methods have been studied using a variational perspective. This has opened up the possibility of introducing variational and symplectic integration methods using geometric integrators. In particular, in thistle, we introduce variational integrators which allows us to derive different methods for optimization. However, since the systems are explicitly time-dependent, the preservation of the symplecticity property occurs solely on the fibers. Finally, using discrete Lagrange-d'Alembert principle we produce optimization methods whose behavior is similar to the classical Nesterov method reducing the oscillations of typical momentum methods. Joint work with Cédric M. Campos and Alejandro Mahillo.

start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: David Martín de Diego (Instituto de Ciencias Matemática, España) - titulo: Sobre sistemas mecánicos discretos forzados y la reducción de Routh discreta abstract: | - Los sistemas lagrangianos y hamiltonianos con fuerzas aparecen en distintos contextos tales como sistemas con fuerzas de control o sistemas con fuerzas de disipación y fricción. También es usual que los sistemas dinámicos que se obtienen al reducir una simetría de un sistema mecánico sean sistemas forzados. En los casos en los que las fuerzas no puedan ser absorbidas por el lagrangiano o el hamiltoniano como parte de un potencial, se realiza una descripción variacional alternativa del caso sin fuerzas. Como en el caso continuo, cuando se realiza un proceso de reducción de una simetría de un sistema mecánico discreto, es usual que el sistema dinámico reducido presentes términos que se pueden interpretar como fuerzas. Entre otras razones, esto hace que resulte interesante estudiar las características de los sistemas discretos forzados. Un caso particularmente interesante de reducción de simetrías de un sistema mecánico es el que se basa en el uso de sus cantidades conservadas. Cuando se considera un sistema mecánico discreto que admite una aplicación momento que se conserva sobre las trayectorias del sistema, se aplica el proceso conocido como la reducción de Routh discreta que da lugar a un sistema forzado. En esta charla se consideran sistemas mecánicos discretos forzados. A partir de su dinámica definida por un principio de Lagrange d'Alembert discreto que se define modificando convenientemente el principio variacional usual, se estudia la existencia de estructuras simplécticas y su posible conservación por la evolución del sistema. En particular, se considera el proceso de reducción de Routh discreta para una simetría no necesariamente abeliana de un sistema mecánico discreto como un caso particular de un proceso de reducción de simetrías más general. También se analizan la existencia y la conservación de estructuras simplécticas sobre el espacio reducido en el marco de los sistemas forzados teniendo en cuenta las características propias de este caso especial. +

Los sistemas lagrangianos y hamiltonianos con fuerzas aparecen en distintos contextos tales como sistemas con fuerzas de control o sistemas con fuerzas de disipación y fricción. También es usual que los sistemas dinámicos que se obtienen al reducir una simetría de un sistema mecánico sean sistemas forzados. En los casos en los que las fuerzas no puedan ser absorbidas por el lagrangiano o el hamiltoniano como parte de un potencial, se realiza una descripción variacional alternativa del caso sin fuerzas. Como en el caso continuo, cuando se realiza un proceso de reducción de una simetría de un sistema mecánico discreto, es usual que el sistema dinámico reducido presentes términos que se pueden interpretar como fuerzas. Entre otras razones, esto hace que resulte interesante estudiar las características de los sistemas discretos forzados. Un caso particularmente interesante de reducción de simetrías de un sistema mecánico es el que se basa en el uso de sus cantidades conservadas. Cuando se considera un sistema mecánico discreto que admite una aplicación momento que se conserva sobre las trayectorias del sistema, se aplica el proceso conocido como la reducción de Routh discreta que da lugar a un sistema forzado. En esta charla se consideran sistemas mecánicos discretos forzados. A partir de su dinámica definida por un principio de Lagrange d'Alembert discreto que se define modificando convenientemente el principio variacional usual, se estudia la existencia de estructuras simplécticas y su posible conservación por la evolución del sistema. En particular, se considera el proceso de reducción de Routh discreta para una simetría no necesariamente abeliana de un sistema mecánico discreto como un caso particular de un proceso de reducción de simetrías más general. También se analizan la existencia y la conservación de estructuras simplécticas sobre el espacio reducido en el marco de los sistemas forzados teniendo en cuenta las características propias de este caso especial.

start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Marcela Zuccalli (Universidad Nacional de La Plata, Argentina) @@ -1026,56 +1026,56 @@ 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. +

It was conjectured by Andruskiewitsch, Angiono and Heckenberger that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system; they did in fact prove this for braidings of affine type or when the rank is two. We shall review some tools developed with the intention of proving this conjecture positively, in a work of Angiono and the author, and exhibit the proof for the rank 3 case, as well as for braidings of Cartan type.

start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Agustín García Iglesias (Universidad Nacional de Córdoba, Argentina) - titulo: A diagrammatic Carlsson-Mellit algebra abstract: | - The \(A_{q,t}\) algebra was introduced by Carlsson and Mellit in their proof of the celebrated shuffle theorem, which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This algebra arises as an extension of two copies of the affine Hecke algebra by certain raising and lowering operators. Carlsson and Mellit constructed an action via plethystic operators on the space of symmetric functions which was then realized geometrically on parabolic flag Hilbert schemes by them and Gorsky. The original algebraic construction was then extended to an infinite family of actions by Mellit and shown to contain the generators of elliptic Hall algebra. However, despite the various formulations of \(A_{q,t}\), performing computations within it is complicated and non-intuitive. - In this talk I will discuss joint work with Matt Hogancamp where we construct a new topological formulation of \(A_{q,t}\) (at t=-1) and its representation as certain braid diagrams on an annulus. In this setting many of the complicated algebraic relations of \(A_{q,t}\) and applications to symmetric functions are trivial consequences of the skein relation imposed on the pictures. In particular, many difficult computations become simple diagrammatic manipulations in this new framework. This purely diagrammatic formulation allows us to lift the operators as certain functors, thus providing a categorification of the \(A_{q,t}\) action on the derived trace of the Soergel category. +

The \(A_{q,t}\) algebra was introduced by Carlsson and Mellit in their proof of the celebrated shuffle theorem, which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This algebra arises as an extension of two copies of the affine Hecke algebra by certain raising and lowering operators. Carlsson and Mellit constructed an action via plethystic operators on the space of symmetric functions which was then realized geometrically on parabolic flag Hilbert schemes by them and Gorsky. The original algebraic construction was then extended to an infinite family of actions by Mellit and shown to contain the generators of elliptic Hall algebra. However, despite the various formulations of \(A_{q,t}\), performing computations within it is complicated and non-intuitive.

+

In this talk I will discuss joint work with Matt Hogancamp where we construct a new topological formulation of \(A_{q,t}\) (at t=-1) and its representation as certain braid diagrams on an annulus. In this setting many of the complicated algebraic relations of \(A_{q,t}\) and applications to symmetric functions are trivial consequences of the skein relation imposed on the pictures. In particular, many difficult computations become simple diagrammatic manipulations in this new framework. This purely diagrammatic formulation allows us to lift the operators as certain functors, thus providing a categorification of the \(A_{q,t}\) action on the derived trace of the Soergel category.

start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Nicolle Gonzalez (The University of California Los Angeles, Estados Unidos) - titulo: Quantum Symmetries in Conformal Field Theory abstract: | - Quantum groups and Nichols algebras appear as symmetries in conformal field theories, as so-called non-local screening operators. I will start by giving a motivation of conformal field theory and explain how this leads from the analysis side to a modular tensor category, quite often one that is well known from algebra. I then explain my recent work that non-local screening operators generate Nichols algebras in this category, and also some recent work with T. Creutzig and M. Rupert that conclude a braided category equivalence between a certain physical model and the representation category of the smallest quantum group \(u_q(\mathfrak{sl}_2),\;q^4=1\). +

Quantum groups and Nichols algebras appear as symmetries in conformal field theories, as so-called non-local screening operators. I will start by giving a motivation of conformal field theory and explain how this leads from the analysis side to a modular tensor category, quite often one that is well known from algebra. I then explain my recent work that non-local screening operators generate Nichols algebras in this category, and also some recent work with T. Creutzig and M. Rupert that conclude a braided category equivalence between a certain physical model and the representation category of the smallest quantum group \(u_q(\mathfrak{sl}_2),\;q^4=1\).

start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Simon Lentner (Universität Hamburg, Alemania) - titulo: Slack Hopf monads abstract: | - Hopf monads are generalizations of Hopf algebras that bring to bear the rich theory of monads from category theory. Even though this generalization overarches many Hopf-like structures, quasi-Hopf algebras seem to be out of its reach. This talk reports on advances in the study of slack Hopf monads, including how these generalize quasi-Hopf algebras. (Joint work with A. Bruguieres and M. Haim). +

Hopf monads are generalizations of Hopf algebras that bring to bear the rich theory of monads from category theory. Even though this generalization overarches many Hopf-like structures, quasi-Hopf algebras seem to be out of its reach. This talk reports on advances in the study of slack Hopf monads, including how these generalize quasi-Hopf algebras. (Joint work with A. Bruguieres and M. Haim).

start: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Ignacio Lopez Franco (Universidad de la República, Uruguay) - titulo: Interpolations of monoidal categories by invariant theory abstract: | - In this talk, I will present a recent construction that enables one to interpolate symmetric monoidal categories by interpolating algebraic structures and their automorphism groups. I will explain how one can recover the constructions of Deligne for categories such as Rep(S_t), Rep(O_t) and Rep(Sp_t), the constructions of Knop for wreath products with S_t and GL_t(O_r), where O_r is a finite quotient of a discrete valuation ring, and also the TQFT categories recently constructed from a rational function by Khovanov, Ostrik, and Kononov. +

In this talk, I will present a recent construction that enables one to interpolate symmetric monoidal categories by interpolating algebraic structures and their automorphism groups. I will explain how one can recover the constructions of Deligne for categories such as Rep(S_t), Rep(O_t) and Rep(Sp_t), the constructions of Knop for wreath products with S_t and GL_t(O_r), where O_r is a finite quotient of a discrete valuation ring, and also the TQFT categories recently constructed from a rational function by Khovanov, Ostrik, and Kononov.

start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Ehud Meir (University of Aberdeen, Escocia) - titulo: Examples of module categories, the non-semisimple case abstract: | - In this talk we present classical examples of exact indecomposable module categories over a finite non-semisimple rigid tensor category, as well as recently constructed new examples. +

In this talk we present classical examples of exact indecomposable module categories over a finite non-semisimple rigid tensor category, as well as recently constructed new examples.

start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: Adriana Mejía Castaño (Universidad del Norte, Colombia) - titulo: Algebras in group-theoretical fusion categories abstract: | - The categorical version of a module over a ring is the notion of a module category over a fusion category. We will first discuss what it means for a module category to be represented by an algebra and introduce the notion of Morita equivalence of algebras in fusion categories. Sonia Natale and Victor Ostrik described algebras representing the Morita equivalence classes in pointed fusion categories. We will explain how this result can be generalized to group-theoretical fusion categories, based on joint work with Monique Müller, Julia Plavnik, Ana Ros Camacho, Angela Tabiri, and Chelsea Walton. +

The categorical version of a module over a ring is the notion of a module category over a fusion category. We will first discuss what it means for a module category to be represented by an algebra and introduce the notion of Morita equivalence of algebras in fusion categories. Sonia Natale and Victor Ostrik described algebras representing the Morita equivalence classes in pointed fusion categories. We will explain how this result can be generalized to group-theoretical fusion categories, based on joint work with Monique Müller, Julia Plavnik, Ana Ros Camacho, Angela Tabiri, and Chelsea Walton.

start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Yiby Morales (Universidad de los Andes, Colombia) - titulo: Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge abstract: | - I will discuss recent work with T. Gannon in which we provide a ribbon tensor equivalence between the representation category of small quantum SL(2) at q=exp(pi*i/p) and the representation category of the triplet VOA at a corresponding central charge 1-6(p-1)^2/p. Such an equivalence was conjectured in work of Gainutdinov, Semikhatov, Tipunin, and Feigin from 2006. +

I will discuss recent work with T. Gannon in which we provide a ribbon tensor equivalence between the representation category of small quantum SL(2) at q=exp(pi*i/p) and the representation category of the triplet VOA at a corresponding central charge 1-6(p-1)^2/p. Such an equivalence was conjectured in work of Gainutdinov, Semikhatov, Tipunin, and Feigin from 2006.

start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Cris Negron (University of North Carolina, Estados Unidos) - titulo: Frobenius-Schur indicators for some families of quadratic fusion categories abstract: | - Quadratic categories are fusion categories with a unique non-trivial orbit from the tensor product action of the group of invertible objects. Familiar examples are the near-groups (with one non-invertible object) and the Haagerup-Izumi cate- gories (with one non-invertible object for each invertible object). Frobenius-Schur indicators are an important invariant of fusion categories generalized from the theory of finite group representations. These indicators may be computed for objects in a fusion category C using the modular data of the Drinfel’d center Z(C) of the fusion category, which is itself a modular tensor category. Recently, Izumi and Grossman provided new (conjectured infinite) families of modular data that include the modular data of Drinfel’d centers for the known quadratic fusion categories. We use this information to compute the FS indicators; moreover, we consider the relationship between the FS indicators of objects in a fusion category C and FS indicators of objects in that category’s Drinfel’d center Z(C). +

Quadratic categories are fusion categories with a unique non-trivial orbit from the tensor product action of the group of invertible objects. Familiar examples are the near-groups (with one non-invertible object) and the Haagerup-Izumi cate- gories (with one non-invertible object for each invertible object). Frobenius-Schur indicators are an important invariant of fusion categories generalized from the theory of finite group representations. These indicators may be computed for objects in a fusion category C using the modular data of the Drinfel’d center Z(C) of the fusion category, which is itself a modular tensor category. Recently, Izumi and Grossman provided new (conjectured infinite) families of modular data that include the modular data of Drinfel’d centers for the known quadratic fusion categories. We use this information to compute the FS indicators; moreover, we consider the relationship between the FS indicators of objects in a fusion category C and FS indicators of objects in that category’s Drinfel’d center Z(C).

start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Henry Tucker (University of California at Riverside, Estados Unidos) @@ -1273,62 +1273,62 @@ 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. +

We adjust the notion of finitary filter pair ( Finitary Filter Pairs and Propositional Logics, South American Journal of Logic 4(2)), which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality \(\kappa\), {where \(\kappa\) is a regular cardinal}. The corresponding new notion is called \(\kappa\)-filter pair. We show that any \(\kappa\)-filter pair gives rise to a logic of cardinality \(\kappa\) and that every logic of cardinality \(\kappa\) comes from a \(\kappa\)-filter pair. We use filter pairs to construct natural extensions for a given logic and work out the relationships between this construction and several others proposed in the literature ( A note on natural extensions in abstract algebraic logic, Studia Logica, 103(4); Constructing natural extensions of propositional logics, Studia Logica 104(6)). Conversely, we describe the class of filter pairs giving rise to a fixed logic in terms of the natural extensions of that logic.

start: 2021-09-15T15:00 end: 2021-09-15T15:45 speaker: Hugo Luiz Mariano (Universidade de São Paulo, Brasil), joint with P. Arndt (University of Düsseldorf, Germany) and D. C. Pinto (Federal University of Bahia, Brazil) - titulo: A topological duality for monotone expansions of semilattices abstract: | - The aim of this work is to present a topological duality for the variety of monotone semilattices. To do this, combining the duality developed by Celani and González for semilattices together with the approach given by Celani and the author for the study of monotone distributive semilattices, we introduce a category of multirelational spaces called \(m\)S-spaces and we prove that this is dually equivalent to the category of monotone semilattices with homomorphisms. Moreover, as an application of this duality we provide a characterization of congruences of (monotone) semilattices by means of lower Vietoris type topologies. It is worth mentioning that a key tool for developing this duality is the topological description of canonical extension of semilattices that we provide. This is a joint work with Ismael Calomino (CIC and Universidad Nacional del Centro) and William J. Zuluaga Botero (Universidad Nacional del Centro). +

The aim of this work is to present a topological duality for the variety of monotone semilattices. To do this, combining the duality developed by Celani and González for semilattices together with the approach given by Celani and the author for the study of monotone distributive semilattices, we introduce a category of multirelational spaces called \(m\)S-spaces and we prove that this is dually equivalent to the category of monotone semilattices with homomorphisms. Moreover, as an application of this duality we provide a characterization of congruences of (monotone) semilattices by means of lower Vietoris type topologies. It is worth mentioning that a key tool for developing this duality is the topological description of canonical extension of semilattices that we provide. This is a joint work with Ismael Calomino (CIC and Universidad Nacional del Centro) and William J. Zuluaga Botero (Universidad Nacional del Centro).

start: 2021-09-15T16:45 end: 2021-09-15T17:30 speaker: María Paula Menchón (Universidad Nacional del Centro de la Provincia de Buenos Aires, Argentina) - titulo: Embeddability between orderings and GCH abstract: | - We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete cardinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure hypothesis for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings. +

We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete cardinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure hypothesis for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings.

start: 2021-09-17T15:45 end: 2021-09-15T16:30 speaker: Rodrigo de Alvarenga Freire (Universidade de Brasília, Brasil) - titulo: Hilbert's tenth problem in rings of meromorphic functions abstract: | - We will give a short survey on definability of the integers in rings of meromorphic functions, together with some ideas on the ingredients in the proofs and the reasons why the case of complex entire functions, as a ring, remain an open problem. We will then present two new evidences for the latter to be undecidable, one obtained with T. Pheidas, and the other one with D. Chompitaki, N. García-Fritz, H. Pasten and T. Pheidas. +

We will give a short survey on definability of the integers in rings of meromorphic functions, together with some ideas on the ingredients in the proofs and the reasons why the case of complex entire functions, as a ring, remain an open problem. We will then present two new evidences for the latter to be undecidable, one obtained with T. Pheidas, and the other one with D. Chompitaki, N. García-Fritz, H. Pasten and T. Pheidas.

start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Xavier Vidaux (Universidad de Concepción, Chile) - titulo: Existentially closed measure preserving actions of the free group. abstract: | - In joint work with Ward Henson and Tomás Ibarlucía, we prove that the theory of atomless probability algebras expanded with a measure preserving action of the free group \(F_k\) has a model companion for any \(k \geq 1\) and the corresponding theory is stable. We build explicitly examples of existentially closed models. Our approach also applies to a reasonably large class of \(\aleph_0\)-categorical superstable theories. +

In joint work with Ward Henson and Tomás Ibarlucía, we prove that the theory of atomless probability algebras expanded with a measure preserving action of the free group \(F_k\) has a model companion for any \(k \geq 1\) and the corresponding theory is stable. We build explicitly examples of existentially closed models. Our approach also applies to a reasonably large class of \(\aleph_0\)-categorical superstable theories.

start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Alexander Berenstein (Universidad de los Andes, Colombia) - titulo: Completitud estándar fuerte (fuerte finita) para lógicas de S5-modales de Łukasiewicz abstract: | - En esta charla mostraremos una versión algebraica de teoremas de completitud estándar fuerte finita para lógicas de S5-modales de Łukasiewicz y algunas de sus extensiones axiomáticas de interés. Para esto mostraremos que la variedad de la MV-álgebras monádicas están generadas como cuasivariedad por cierta álgebra funcional estándar. Lo mismo se hará para ciertas extensiones axiomáticas de interés. Por último agregaremos una regla infinitaria y mostraremos la completitud estándar fuerte para esa lógica usando nuevamente técnicas algebraicas. +

En esta charla mostraremos una versión algebraica de teoremas de completitud estándar fuerte finita para lógicas de S5-modales de Łukasiewicz y algunas de sus extensiones axiomáticas de interés. Para esto mostraremos que la variedad de la MV-álgebras monádicas están generadas como cuasivariedad por cierta álgebra funcional estándar. Lo mismo se hará para ciertas extensiones axiomáticas de interés. Por último agregaremos una regla infinitaria y mostraremos la completitud estándar fuerte para esa lógica usando nuevamente técnicas algebraicas.

start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: José Patricio Díaz Varela (Universidad Nacional del Sur, Argentina) - titulo: Distributivity, Modularity, and Natural Deduction abstract: | - On the one hand, from a logical point of view, we will pay attention to discussions around the notion of distributivity in the nineteenth century world of the algebra of logic. In particular, we will consider some passages from the writings of Ernst Schršoder and Charles Peirce. On the other hand, from an algebraic point of view, we will trace the origins of the notion of modularity in lattice theory. This we will do examining some writings of Richard Dedekind. Moreover, we will 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. +

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: 2021-09-15T15:45 end: 2021-09-15T16:30 speaker: Rodolfo C. Ertola-Biraben (Universidad de Campinas, Brasil) - titulo: Interpolation and Beth definability for conic idempotent Full Lambek calculus abstract: | - The Full Lambek calculus forms the basic system for substructural logics, examples of which include relevance, linear, many-valued, intuitionistic and classical logic. We focus on substructural logics that satisfy contraction, mingle and the rule of conicity. - The algebraic semantics of substructural logics are residuated lattices and they have an independent history in the context of order algebra; they were first introduced by Ward and Dilworth as tools in the study of ideal lattices of rings. Residuated lattices have a monoid and a lattice reduct, as well as division-like operations; examples include Boolean algebras, lattice-ordered groups and relation algebras. They are also connected to mathematical linguistics and computer science (for example pointer management and memory allocation). We focus on idempotent conic residuated lattices, which are related to algebraic models of relevance logic, and which extend the class of idempotent linear ones. After establishing a decomposition result for this class, we show that it has the strong amalgamation property, and extend the result to the variety generated by this class; this implies that the corresponding logic has the interpolation property and the Beth definability property. +

The Full Lambek calculus forms the basic system for substructural logics, examples of which include relevance, linear, many-valued, intuitionistic and classical logic. We focus on substructural logics that satisfy contraction, mingle and the rule of conicity.

+

The algebraic semantics of substructural logics are residuated lattices and they have an independent history in the context of order algebra; they were first introduced by Ward and Dilworth as tools in the study of ideal lattices of rings. Residuated lattices have a monoid and a lattice reduct, as well as division-like operations; examples include Boolean algebras, lattice-ordered groups and relation algebras. They are also connected to mathematical linguistics and computer science (for example pointer management and memory allocation). We focus on idempotent conic residuated lattices, which are related to algebraic models of relevance logic, and which extend the class of idempotent linear ones. After establishing a decomposition result for this class, we show that it has the strong amalgamation property, and extend the result to the variety generated by this class; this implies that the corresponding logic has the interpolation property and the Beth definability property.

start: 2021-09-15T17:30 end: 2021-09-15T18:15 speaker: Nick Galatos (University of Denver, Estados Unidos), joint with Wesley Fussner - titulo: Lindström theorems in graded model theory abstract: | - Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this talk I will present some first steps towards an abstract formulation of this model theory. I will give a general notion of abstract logic based on many-valued models and discuss six Lindstr\"{o}m-style characterizations of maximality of first-order logics in terms of metalogical properties such as compactness, abstract completeness, the L\"{o}wenheim-Skolem property, the Tarski union property, and the Robinson property, among others. The results have been obtained in a joint work with Guillermo Badia. +

Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this talk I will present some first steps towards an abstract formulation of this model theory. I will give a general notion of abstract logic based on many-valued models and discuss six Lindstr\"{o}m-style characterizations of maximality of first-order logics in terms of metalogical properties such as compactness, abstract completeness, the L\"{o}wenheim-Skolem property, the Tarski union property, and the Robinson property, among others. The results have been obtained in a joint work with Guillermo Badia.

start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Carles Noguera i Clofent (Czech Academy of Sciences, República Checa) - titulo: Locally pseudocomplemented abelian \(\ell\)-groups abstract: | - It is a known result of Glass and Pierce that the variety of abelian lattice ordered groups (\(\ell\)-groups) does not to have a model companion. Call and abelian \(\ell\)-group\ \emph{locally pseudocomplemented} if any bounded lattice interval in the group is pseudocomplemented. These groups form a discriminator variety which has a model completion, namely the divisible locally pseudocomplemented abelian \(\ell\)-groups with dense boolean part. We have similar results for pseudocomplemented MV-algebras, the intervals of locally pseudocomplemented abelian \(\ell\)-groups, actually Heyting (Gödel) algebras and thus the algebraic semantics for the union of Łukasiewicz and Gödel-Dummett logic. Some applications follow on definability in these varieties and the corresponding logics (an expansion of abelian logic in the former case). These results are related to recent work by Metcalfe and Reggio. +

It is a known result of Glass and Pierce that the variety of abelian lattice ordered groups (\(\ell\)-groups) does not to have a model companion. Call and abelian \(\ell\)-group\ \emph{locally pseudocomplemented} if any bounded lattice interval in the group is pseudocomplemented. These groups form a discriminator variety which has a model completion, namely the divisible locally pseudocomplemented abelian \(\ell\)-groups with dense boolean part. We have similar results for pseudocomplemented MV-algebras, the intervals of locally pseudocomplemented abelian \(\ell\)-groups, actually Heyting (Gödel) algebras and thus the algebraic semantics for the union of Łukasiewicz and Gödel-Dummett logic. Some applications follow on definability in these varieties and the corresponding logics (an expansion of abelian logic in the former case). These results are related to recent work by Metcalfe and Reggio.

start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Xavier Caicedo (Universidad de los Andes, Colombia) @@ -1343,52 +1343,52 @@ 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. +

A separated graph is a pair \((E,C)\), where \(E\) is a directed graph, \(C=\bigsqcup _{v\in E^ 0} C_v\), and \(C_v\) is a partition of \(r^{-1}(v)\) (into pairwise disjoint nonempty subsets) for every vertex \(v\). In recent years, separated graphs have been used to provide combinatorial models of several structures, often related to dynamical systems. This can be understood as a generalization of the common use of usual directed graphs in symbolic dynamics. I will survey some of these developments, including the failure of Tarski's dichotomy in the setting of topological actions, the construction of a family of ample groupoids with prescribed type semigroup, and the modeling of actions on the Cantor set.

start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Pere Ara (Universitat Autònoma de Barcelona, España) - titulo: A non-commutative topological dimension abstract: | - Due to the Gelfand duality, C*-algebras are regarded as non-commutative topological spaces. This approach has sparked fundamental ideas in the structure and classification theory of C*-algebras. In this talk, I will explain a notion of non-commutative covering dimension and its impact in the classification programme of C*-algebras. +

Due to the Gelfand duality, C*-algebras are regarded as non-commutative topological spaces. This approach has sparked fundamental ideas in the structure and classification theory of C*-algebras. In this talk, I will explain a notion of non-commutative covering dimension and its impact in the classification programme of C*-algebras.

start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Jorge Castillejos (Instituto de Ciencias Matemáticas ICMAT, España) - titulo: Isotropy algebras and Fourier analysis for regular inclusions abstract: | - Given a regular inclusion \(A\subseteq B\) of C*-algebras, with \(A\) abelian, we will define the notion of ``isotropy algebra'' for any given point \(x\) in the spectrum of \(A.\) We will then compare this notion with the usual notion of isotropy groups in the context of groupoid C*-algebras, showing that the isotropy algebra relative to the full groupoid C*-algebra is naturally isomorphic to the full group C*-algebra of the corresponding isotropy group. Based on the slightly generalized notion of ``isotropy module'', we will then introduce a method to analyze the structure of \(B\) paralleling the classical Fourier analysis. This talk is based on the paper [Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids, arXiv:1901.09683 [math.OA] [v3] Tue, 1 Jun 2021], joint with David Pitts. +

Given a regular inclusion \(A\subseteq B\) of C*-algebras, with \(A\) abelian, we will define the notion of ``isotropy algebra'' for any given point \(x\) in the spectrum of \(A.\) We will then compare this notion with the usual notion of isotropy groups in the context of groupoid C*-algebras, showing that the isotropy algebra relative to the full groupoid C*-algebra is naturally isomorphic to the full group C*-algebra of the corresponding isotropy group. Based on the slightly generalized notion of ``isotropy module'', we will then introduce a method to analyze the structure of \(B\) paralleling the classical Fourier analysis. This talk is based on the paper [Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids, arXiv:1901.09683 [math.OA] [v3] Tue, 1 Jun 2021], joint with David Pitts.

start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Ruy Exel (Universidade Federal de Santa Catarina, Brasil) - titulo: Toeplitz algebras of semigroups and their boundary quotients abstract: | - In general there are many distinct C*-algebras generated by isometric representations of a left cancellative monoid \(P\). Two distinguished examples are the Toeplitz C*-algebra \(\mathcal{T}_\lambda(P)\) generated by the left regular representation of \(P\) on \(\ell^2(P)\), and its boundary quotient \(\partial\mathcal{T}_\lambda(P)\). I will report on recent work with M. Laca, in which we define a universal Toeplitz C*-algebra \(\mathcal{T}_u(P)\) for a submonoid of a group that is canonically isomorphic to Li's semigroup C*-algebra when independence holds and is the universal analogue of \(\mathcal{T}_\lambda(P)\) also when independence fails. I plan to focus mainly on boundary quotients, explaining why the covariance algebra of the canonical product system over \(P\) with one-dimensional fibres may be regarded as the universal analogue of \(\partial\mathcal{T}_\lambda(P)\). I will give a concrete presentation of the full boundary quotient using a new notion of foundation sets, and give sufficient conditions on \(P\) for \(\partial\mathcal{T}_\lambda(P)\) to be purely infinite simple. If time permits, I will also address faithfulness of representations at the level of Toeplitz algebras. +

In general there are many distinct C*-algebras generated by isometric representations of a left cancellative monoid \(P\). Two distinguished examples are the Toeplitz C*-algebra \(\mathcal{T}_\lambda(P)\) generated by the left regular representation of \(P\) on \(\ell^2(P)\), and its boundary quotient \(\partial\mathcal{T}_\lambda(P)\). I will report on recent work with M. Laca, in which we define a universal Toeplitz C*-algebra \(\mathcal{T}_u(P)\) for a submonoid of a group that is canonically isomorphic to Li's semigroup C*-algebra when independence holds and is the universal analogue of \(\mathcal{T}_\lambda(P)\) also when independence fails. I plan to focus mainly on boundary quotients, explaining why the covariance algebra of the canonical product system over \(P\) with one-dimensional fibres may be regarded as the universal analogue of \(\partial\mathcal{T}_\lambda(P)\). I will give a concrete presentation of the full boundary quotient using a new notion of foundation sets, and give sufficient conditions on \(P\) for \(\partial\mathcal{T}_\lambda(P)\) to be purely infinite simple. If time permits, I will also address faithfulness of representations at the level of Toeplitz algebras.

start: 2021-09-16T17:30 end: 2021-09-16T18:15 speaker: Camila Senhem (Victoria University of Wellington, Nueva Zelanda) - titulo: Lifts of completely positive maps abstract: | - Let \(A\) and \(B\) be \(C^*\)-algebras, \(A\) separable and \(I\) an ideal in \(B\). We show that for any completely positive contractive linear map \(\psi\colon A\to B/I\) there is a continuous family \(\Theta_t\colon A\to B\), for \(t\in [1,\infty)\), of lifts of \(\psi\) that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If \(A\) and \(B\) carry continuous actions of a second countable locally compact group \(G\) such that \(I\) is \(G\)-invariant and \(\psi\) is equivariant, then the family \(\Theta_t\) can be chosen to be asymptotically equivariant. - If a linear completely positive lift for \(\psi\) exists, then we can arrange that \(\Theta_t\) is linear and completely positive for all \(t\in [1,\infty)\); this yields an equivariant version of the Choi-Effros lifting theorem. In the equivariant setting, if \(A\), \(B\) and \(\psi\) are unital, the existence of asymptotically linear unital lifts are only guaranteed if \(G\) is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps. - This talk is based on joint work with Marzieh Forough and Klaus Thomsen. +

Let \(A\) and \(B\) be \(C^*\)-algebras, \(A\) separable and \(I\) an ideal in \(B\). We show that for any completely positive contractive linear map \(\psi\colon A\to B/I\) there is a continuous family \(\Theta_t\colon A\to B\), for \(t\in [1,\infty)\), of lifts of \(\psi\) that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If \(A\) and \(B\) carry continuous actions of a second countable locally compact group \(G\) such that \(I\) is \(G\)-invariant and \(\psi\) is equivariant, then the family \(\Theta_t\) can be chosen to be asymptotically equivariant.

+

If a linear completely positive lift for \(\psi\) exists, then we can arrange that \(\Theta_t\) is linear and completely positive for all \(t\in [1,\infty)\); this yields an equivariant version of the Choi-Effros lifting theorem. In the equivariant setting, if \(A\), \(B\) and \(\psi\) are unital, the existence of asymptotically linear unital lifts are only guaranteed if \(G\) is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.

+

This talk is based on joint work with Marzieh Forough and Klaus Thomsen.

start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Eusebio Gardella (Universidad de Münster, Alemania) - titulo: Continuous orbit equivalence and full groups of ultragraph C*-algebras abstract: | - In this talk, we describe ultragraphs, their associated edge shift spaces (which generalize SFT for infinite alphabets), and their associated C*-algebras and groupoids. We present results regarding continuous orbit equivalence of Deaconu-Renault systems and full groups associated with groupoids. To finish, we describe how to apply these results for ultragraph algebras. +

In this talk, we describe ultragraphs, their associated edge shift spaces (which generalize SFT for infinite alphabets), and their associated C*-algebras and groupoids. We present results regarding continuous orbit equivalence of Deaconu-Renault systems and full groups associated with groupoids. To finish, we describe how to apply these results for ultragraph algebras.

start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Daniel Gonçalves (Universidade Federal de Santa Catarina, Brasil) - titulo: Revisiting quantum mechanics via quantales and groupoid C*-algebras abstract: | - It is well known that operator algebras are the mathematical language of algebraic quantum mechanics and algebraic quantum field theory. In particular, von Neumann algebras cater for a natural formalization of Heisenberg's ``picture'' of quantum mechanics, which rests on interpreting physical observables as being time-dependent self-adjoint operators, to which von Neumann explicitly added, still in his twenties, that physical measurements should correspond to projection operators, thus producing the first mathematically thorough account of the foundations of quantum theory. However, from a conceptual point of view deep problems remained. These, in essence, have led to the seemingly unstoppable proliferation of interpretations and modifications of quantum mechanics, despite which, to this day, the ``measurement problem'' remains largely unsolved. In this talk I describe ongoing work whose aim is to address this by means of a definition of space of measurements whose open sets correspond to the finite pieces of classical information that can be extracted during a measurement. The interplay between C*-algebras, locally compact étale groupoids, and quantales plays an important role. The main purpose of the talk is to explain the main ideas surrounding measurement spaces along with related facts about groupoid C*-algebras and Fell bundles. +

It is well known that operator algebras are the mathematical language of algebraic quantum mechanics and algebraic quantum field theory. In particular, von Neumann algebras cater for a natural formalization of Heisenberg's ``picture'' of quantum mechanics, which rests on interpreting physical observables as being time-dependent self-adjoint operators, to which von Neumann explicitly added, still in his twenties, that physical measurements should correspond to projection operators, thus producing the first mathematically thorough account of the foundations of quantum theory. However, from a conceptual point of view deep problems remained. These, in essence, have led to the seemingly unstoppable proliferation of interpretations and modifications of quantum mechanics, despite which, to this day, the ``measurement problem'' remains largely unsolved. In this talk I describe ongoing work whose aim is to address this by means of a definition of space of measurements whose open sets correspond to the finite pieces of classical information that can be extracted during a measurement. The interplay between C*-algebras, locally compact étale groupoids, and quantales plays an important role. The main purpose of the talk is to explain the main ideas surrounding measurement spaces along with related facts about groupoid C*-algebras and Fell bundles.

start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Pedro Resende (Instituto Superior Técnico, Portugal) - titulo: Permanence properties of verbal products and verbal wreath products of groups abstract: | - Given a family of groups, the direct sum and the free product provide useful ways of constructing new groups out of it. Verbal products of groups, which were defined in the fifties, give many new operations in groups which share some common features with the direct sum and the free product. Arguably, these products have not been studied much in recent years and they might have been overlooked in geometric and measurable group theory. - In this talk we will review these products, give some examples and show that several group theoretical properties that are of much interest in operator algebras like amenability, Haagerup property, property (T), exactness, soficity, Connes' embedability, etc. are preserved under some of these products. Afterwards, we will explain how to define verbal wreath products between two groups, (these are semi-direct products that resemble the restricted wreath product), and we will show that the Haagerup property, soficity, Connes' embeddability and other metric approximation properties are preserved under taking verbal wreath products. Finally we will give some applications of them. +

Given a family of groups, the direct sum and the free product provide useful ways of constructing new groups out of it. Verbal products of groups, which were defined in the fifties, give many new operations in groups which share some common features with the direct sum and the free product. Arguably, these products have not been studied much in recent years and they might have been overlooked in geometric and measurable group theory.

+

In this talk we will review these products, give some examples and show that several group theoretical properties that are of much interest in operator algebras like amenability, Haagerup property, property (T), exactness, soficity, Connes' embedability, etc. are preserved under some of these products. Afterwards, we will explain how to define verbal wreath products between two groups, (these are semi-direct products that resemble the restricted wreath product), and we will show that the Haagerup property, soficity, Connes' embeddability and other metric approximation properties are preserved under taking verbal wreath products. Finally we will give some applications of them.

start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Román Sasyk (Universidad de Buenos Aires, Argentina) @@ -1594,39 +1594,39 @@ 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. +

Transmission problems originally arose in elasticity theory, and they are nowadays of great interest due to its many applications in different areas in science. Typically, in these problems, there is a fixed interface where solutions may change abruptly, and the primary focus is to study their behavior across this surface. In this talk, we will discuss existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with \(C^{1,\alpha}\) interfaces. For this, we develop a novel geometric stability argument based on the mean value property.

+

These results are part of my PhD dissertation, and they are joint work with Luis A. Caffarelli and Pablo R. Stinga.

start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: María Soria-Carro (University of Texas, Estados Unidos) - titulo: A Non-variational Approach for the Porous Medium Equation abstract: | - We extend the Krylov-Safonov theory and the corresponding H\"older estimates for the viscosity solutions of a type porous medium equation with non-variational structure. This work is in collaboration with Makson Santos (CIMAT-Guanajuato). +

We extend the Krylov-Safonov theory and the corresponding H\"older estimates for the viscosity solutions of a type porous medium equation with non-variational structure. This work is in collaboration with Makson Santos (CIMAT-Guanajuato).

start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Héctor Andrés Chang Lara (Centro de Investigación en Matemáticas, México) - titulo: Blow up for the Keller-Segel system in the critical mass case abstract: | - We consider the Keller-Segel system in the plane with an initial condition with suitable decay and mass \(8 \pi\), which corresponds to the threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function \(u_0\) with mass \(8 \pi\) such that for any initial condition sufficiently close to \(u_0\) and the same mass, the solution is globally defined and blows-up in infinite time. We also find the profile and rate of blow-up. This result answers affirmatively the question of the nonradial stability raised by Ghoul and Masmoudi (2018). This is a joint work with M. del Pino, M. Musso and J. Wei. +

We consider the Keller-Segel system in the plane with an initial condition with suitable decay and mass \(8 \pi\), which corresponds to the threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function \(u_0\) with mass \(8 \pi\) such that for any initial condition sufficiently close to \(u_0\) and the same mass, the solution is globally defined and blows-up in infinite time. We also find the profile and rate of blow-up. This result answers affirmatively the question of the nonradial stability raised by Ghoul and Masmoudi (2018). This is a joint work with M. del Pino, M. Musso and J. Wei.

start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Juan Dávila (University of Bath, Inglaterra y Universidad de Antioquia, Colombia) - titulo: Critical polyharmonic systems and optimal partitions abstract: | - In this talk, we establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in \(\mathbb{R}^N\) which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to \(-\infty\). We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of \(\mathbb{R}^N\). Moreover, we give a detailed description of the shape of these domains. - This is a joint work with Mónica Clapp and Alberto Saldaña. +

In this talk, we establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in \(\mathbb{R}^N\) which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to \(-\infty\). We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of \(\mathbb{R}^N\). Moreover, we give a detailed description of the shape of these domains.

+

This is a joint work with Mónica Clapp and Alberto Saldaña.

start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Juan Carlos Fernández Morelos (Universidad Nacional Autónoma de México, México) - titulo: Blowing up solutions for sinh-Poisson type equations on pierced domains abstract: | - We consider sinh-Poisson type equations (either Liouville or mean field form) with variable intensities and Dirichlet boundary condition on a pierced domain \(\Omega_\varepsilon:=\Omega\setminus \cup_{i=1}^m B(\xi_i,\varepsilon_i)\), where \(\Omega\) is a smooth bounded domain in \(\text{I\!R}^2\) which contains the points \(\xi_i\), \(i=1,\dots,m\), \(\varepsilon_i=\varepsilon_i(\varepsilon)>0\), \(i=1,\dots,m\), depending on some parameter \(\varepsilon>0\) small enough. Given \(m\) different points \(\xi_i\), \(i=1,\dots,m\), we have found suitable radii \(\varepsilon_i\), \(i=1,\dots,m\) such that for \(\varepsilon>0\) small enough our problem (either Liouville or mean field form) has a solution \(u_\varepsilon\) blowing up positively around each \(\xi_1,\dots,\xi_{m_1}\) and negatively around \(\xi_{m_1+1},\dots,\xi_m\) respectively, as \(\varepsilon\to0\). We have used a family of solutions of the singular Liouville equation +

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). + to construct an approximation of the solution depending on some parameters, suitable projected and scaled in order to make the error small enough in a suitable norm. Hence, we have found an actual solution as a small additive perturbation of this initial approximation by using a fixed point argument.

+

Part of these results have been obtained in collaboration with Pierpaolo Esposito (Università di Roma Tre, Italia) and Angela Pistoia (Università di Roma ``La Sapienza'', Italia).

start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Pablo Figueroa (Universidad Austral de Chile, Chile) @@ -1634,13 +1634,13 @@ 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). +

Joint work with P. Blanc (Jyväskylä), F. Charro (Detroit), and J.J. Manfredi (Pittsburgh).

start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Julio D. Rossi (Universidad de Buenos Aires, Argentina) - titulo: Radially symmetric solutions to quasilinear problems with indefinite weight abstract: | - In this talk we present recent results regarding the existence of infinitely many radially symmetric solutions to +

In this talk we present recent results regarding the existence of infinitely many radially symmetric solutions to \[\label{eq main problem} \left\{ \begin{aligned} @@ -1649,19 +1649,19 @@ \end{aligned} \right. \] - 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. + where \(N\geq 2\), \(p>1\), \(\Delta _p u= \text{div} (|\nabla u|^{p-2} \nabla u)\) is the p-Laplacian operator, \(B_1 (0)\) stands for the unit ball in \(\mathbb{R} ^N\) centered at the origin, \(g: \mathbb{R} \longrightarrow \mathbb{R} \) is a nonlinear function which satisfies p-superlinearity conditions and the weight \(W: [0,1]\longrightarrow \mathbb{R}\) is a \(C^1 -\)function that changes sign.

+

Since the weight \(W\) takes negative values, the solutions to initial value problems associated to eq main problem may blow up at some \(r \in (X,1)\) and this fact represents a major difficulty when using phase-plane analysis arguments. We will discuss how this difficulty can be surpassed.

+

This is a recent work in collaboration with Alfonso Castro, Jorge Cossio and Sigifredo Herrón.

start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Carlos Vélez (Universidad Nacional de Colombia, Colombia) - titulo: 'A nonlocal isoperimetric problem: density perimeter' abstract: | - We will discuss a variant of a classical geometric minimization problem in arbitrary dimensions \(d\geq 2\), known as nonlocal isoperimetric problem, which arises from studies in Nuclear Physics by Gamow in the 1930’s: +

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. + for any \(\alpha\in(0,d)\) and \(\gamma>0\). By introducing a density \(a:\mathbb{R}^d\to[0,\infty)\) in the perimeter functional, we obtain features that differ substantially from existing results in the framework of the classical problem without densities. Firstly, we get existence of minimizing sets for a general class of coercive densities \(a(x)\) and all values of ``mass". In addition, we prove essential boundedness of such minimizers. Finally, in the regime of “small” non-local contribution we completely characterize the minimizer, for densities that are monomial radial weights. This work is a collaboration with Stan Alama and Lia Bronsard (McMaster University) and Ihsan Topaloglu (Virginia Commonwealth University), as part of the project QUALITATIVE PROPERTIES OF WEIGHTED AND ANISOTROPIC VARIATIONAL PROBLEMS financed by ANID CHILE FONDECYT INICIACION No. 11201259.

start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Andres Zuñiga (Universidad de O’Higgins, Chile) @@ -1738,53 +1738,53 @@ 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. +

In this seminar I will describe the derivation of certain laws that rule the space-time evolution of the conserved quantities of stochastic processes. The random dynamics conserves a quantity (as the total mass) that has a non-trivial evolution in space and time. The goal is to describe the connection between the macroscopic (continuous) equations and the microscopic (discrete) system of random particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling; while the latter is a collection of particles that move randomly according to a transition probability and rings of Poisson processes. I will focus on a model for which we can obtain a collection of (fractional) reaction-diffusion equations given in terms of the regional fractional Laplacian with different types of boundary conditions.

start: 2021-09-16T15:00 end: 2021-09-16T15:45 speaker: Patrícia Gonçalves (Instituto Superior Técnico, Portugal) - titulo: Persistence phenomena for large biological neural networks abstract: | - We study a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of biological mechanisms. We focus here on local interactions: upon firing, a given neuron increases the potential of a fixed number of randomly chosen neurons. We show a phase transition in terms of the stationary distribution of the limiting network. Whereas a finite network activity always vanishes in finite time, the infinite network might converge to either a trivial stationary measure or to a nontrivial one. This allows to model the biological phenomena of persistence: we prove that the network may retain neural activity for large times depending on certain global parameters describing the intensity of interconnection. - Joint work with: Maximiliano Altamirano, Roberto Cortez, Lasse Leskela. +

We study a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of biological mechanisms. We focus here on local interactions: upon firing, a given neuron increases the potential of a fixed number of randomly chosen neurons. We show a phase transition in terms of the stationary distribution of the limiting network. Whereas a finite network activity always vanishes in finite time, the infinite network might converge to either a trivial stationary measure or to a nontrivial one. This allows to model the biological phenomena of persistence: we prove that the network may retain neural activity for large times depending on certain global parameters describing the intensity of interconnection.

+

Joint work with: Maximiliano Altamirano, Roberto Cortez, Lasse Leskela.

start: 2021-09-17T15:00 end: 2021-09-17T15:45 speaker: Matthieu Jonckheere (Universidad de Buenos Aires, Argentina) - titulo: Structure recovery for partially observed discrete Markov random fields on graphs abstract: | - Discrete Markov random fields on graphs, also known as graphical models in the statistical literature, have become popular in recent years due to their flexibility to capture conditional dependency relationships between variables. They have already been applied to many different problems in different fields such as Biology, Social Science, or Neuroscience. Graphical models are, in a sense, finite versions of general random fields or Gibbs distributions, classical models in stochastic processes. This talk will present the problem of estimating the interaction structure (conditional dependencies) between variables by a penalized pseudo-likelihood criterion. First, I will consider this criterion to estimate the interaction neighborhood of a single node, which will later be combined with the other estimated neighborhoods to obtain an estimator of the underlying graph. I will show some recent consistency results for the estimated neighborhood of a node and any finite sub-graph when the number of candidate nodes grows with the sample size. These results do not assume the usual positivity condition for the conditional probabilities of the model as it is usually assumed in the literature of Markov random fields. These results open new possibilities of extending these models to situations with sparsity, where many parameters of the model are null. I will also present some ongoing extensions of these results to processes satisfying mixing type conditions. This is joint work with Iara Frondana, Rodrigo Carvalho and Magno Severino. +

Discrete Markov random fields on graphs, also known as graphical models in the statistical literature, have become popular in recent years due to their flexibility to capture conditional dependency relationships between variables. They have already been applied to many different problems in different fields such as Biology, Social Science, or Neuroscience. Graphical models are, in a sense, finite versions of general random fields or Gibbs distributions, classical models in stochastic processes. This talk will present the problem of estimating the interaction structure (conditional dependencies) between variables by a penalized pseudo-likelihood criterion. First, I will consider this criterion to estimate the interaction neighborhood of a single node, which will later be combined with the other estimated neighborhoods to obtain an estimator of the underlying graph. I will show some recent consistency results for the estimated neighborhood of a node and any finite sub-graph when the number of candidate nodes grows with the sample size. These results do not assume the usual positivity condition for the conditional probabilities of the model as it is usually assumed in the literature of Markov random fields. These results open new possibilities of extending these models to situations with sparsity, where many parameters of the model are null. I will also present some ongoing extensions of these results to processes satisfying mixing type conditions. This is joint work with Iara Frondana, Rodrigo Carvalho and Magno Severino.

start: 2021-09-16T15:45 end: 2021-09-16T16:30 speaker: Florencia Leonardi (Universidade de São Paulo, Brasil) - titulo: An algorithm to solve optimal stopping problems for one-dimensional diffusions abstract: | - Considering a real-valued diffusion, a real-valued reward function and a positive discount rate, we provide an algorithm to solve the optimal stopping problem consisting in finding the optimal expected discounted reward and the optimal stopping time at which it is attained. Our approach is based on Dynkin's characterization of the value function. The combination of Riesz's representation of r-excessive functions and the inversion formula gives the density of the representing measure, being only necessary to determine its support. This last task is accomplished through an algorithm. The proposed method always arrives to the solution, thus no verification is needed, giving, in particular, the shape of the stopping region. Generalizations to diffusions with atoms in the speed measure and to non smooth payoffs are analyzed. +

Considering a real-valued diffusion, a real-valued reward function and a positive discount rate, we provide an algorithm to solve the optimal stopping problem consisting in finding the optimal expected discounted reward and the optimal stopping time at which it is attained. Our approach is based on Dynkin's characterization of the value function. The combination of Riesz's representation of r-excessive functions and the inversion formula gives the density of the representing measure, being only necessary to determine its support. This last task is accomplished through an algorithm. The proposed method always arrives to the solution, thus no verification is needed, giving, in particular, the shape of the stopping region. Generalizations to diffusions with atoms in the speed measure and to non smooth payoffs are analyzed.

start: 2021-09-17T15:45 end: 2021-09-17T16:30 speaker: Ernesto Mordecki (Universidad de la República, Uruguay) - titulo: Branching processes with pairwise interactions abstract: | - In this talk, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. BPI-processes form a subclass of branching processes with interactions, which were recently introduced by González Casanova et al. (2021), and includes the so-called logistic branching process which was studied by Lambert (2005). - Here, we provide a series of integral tests that fully explains how competition and cooperation regulates the long-term behaviour of BPI-processes. In particular, we give necessary and sufficient conditions for the events of explosion and extinction, as well as conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use the moment dual of BPI-processes which turns out to be a family of diffusions taking values on \([0,1]\), that we introduce as generalised Wright-Fisher diffusions together with a complete understanding of the nature of their boundaries. +

In this talk, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. BPI-processes form a subclass of branching processes with interactions, which were recently introduced by González Casanova et al. (2021), and includes the so-called logistic branching process which was studied by Lambert (2005).

+

Here, we provide a series of integral tests that fully explains how competition and cooperation regulates the long-term behaviour of BPI-processes. In particular, we give necessary and sufficient conditions for the events of explosion and extinction, as well as conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use the moment dual of BPI-processes which turns out to be a family of diffusions taking values on \([0,1]\), that we introduce as generalised Wright-Fisher diffusions together with a complete understanding of the nature of their boundaries.

start: 2021-09-16T17:30 end: 2021-09-17T18:15 speaker: Juan Carlos Pardo (Universidad Nacional Autónoma de México, México) - titulo: Exact solution of TASEP and generalizations abstract: | - I will present a general result which allows to express the multipoint distribution of the particle locations in the totally asymmetric exclusion process (TASEP) and several related processes, for general initial conditions, in terms of the Fredholm determinant of certain kernels involving the hitting time of a random walk to a curve defined by the initial data. This scheme generalizes an earlier result for the particular case of continuous time TASEP, which has been used to prove convergence of TASEP to the KPZ fixed point. The result covers processes in continuous and discrete time, with push and block dynamics, as well as some extensions to processes with memory length larger than 1. - Based on joint work with Konstantin Matetski. +

I will present a general result which allows to express the multipoint distribution of the particle locations in the totally asymmetric exclusion process (TASEP) and several related processes, for general initial conditions, in terms of the Fredholm determinant of certain kernels involving the hitting time of a random walk to a curve defined by the initial data. This scheme generalizes an earlier result for the particular case of continuous time TASEP, which has been used to prove convergence of TASEP to the KPZ fixed point. The result covers processes in continuous and discrete time, with push and block dynamics, as well as some extensions to processes with memory length larger than 1.

+

Based on joint work with Konstantin Matetski.

start: 2021-09-17T16:45 end: 2021-09-17T17:30 speaker: Daniel Remenik (Universidad de Chile, Chile) - titulo: Percolation on Randomly Stretched Lattices abstract: | - In this talk we will give a new proof for a question that was first posed by Jonasson, Mossel and Peres, concerning percolation on a randomly stretched planar lattice. More specifically, we fix a parameter q in (0, 1) and we slash the lattice Z^2 in the following way. For every vertical line that crosses the x axis along an integer value, we toss an independent coin and with probability q we remove all edges along that line. Then we do the same with the horizontal lines that cross the y axis at integer values. We are then left with a graph G that looks like a randomly stretched version of Z^2 and on top of which we would like to perform i.i.d. Bernoulli percolation. The question at hand is whether this percolation features an non-trivial phase transition, or more precisely, whether p_c(G) < 1. Although this question has been previously solved in a seminal article by Hoffman, we present here an alternative solution that greatly simplifies the exposition. We also explain how the presented techniques can be used to prove the existence of a phase transition for other models with minimal changes to the proof. - This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis. +

In this talk we will give a new proof for a question that was first posed by Jonasson, Mossel and Peres, concerning percolation on a randomly stretched planar lattice. More specifically, we fix a parameter q in (0, 1) and we slash the lattice Z^2 in the following way. For every vertical line that crosses the x axis along an integer value, we toss an independent coin and with probability q we remove all edges along that line. Then we do the same with the horizontal lines that cross the y axis at integer values. We are then left with a graph G that looks like a randomly stretched version of Z^2 and on top of which we would like to perform i.i.d. Bernoulli percolation. The question at hand is whether this percolation features an non-trivial phase transition, or more precisely, whether p_c(G) < 1. Although this question has been previously solved in a seminal article by Hoffman, we present here an alternative solution that greatly simplifies the exposition. We also explain how the presented techniques can be used to prove the existence of a phase transition for other models with minimal changes to the proof.

+

This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis.

start: 2021-09-16T16:45 end: 2021-09-16T17:30 speaker: Augusto Teixeira (Instituto de Matemática Pura e Aplicada, Brasil) - titulo: A martingale approach to lumpability abstract: | - The martingale problem introduced by Stroock and Varadhan is an efficient method to prove the convergence of a sequence of stochastic processes which are derived from Markov processes. In this talk we present two examples to illustrate this approach: the density of particles per site of a sequence of condensing zero range processes and the number of sites occupied by a system of coalescing random walks evolving on a transitive finite graph. Both examples exhibit a sort of asymptotic lumpability. +

The martingale problem introduced by Stroock and Varadhan is an efficient method to prove the convergence of a sequence of stochastic processes which are derived from Markov processes. In this talk we present two examples to illustrate this approach: the density of particles per site of a sequence of condensing zero range processes and the number of sites occupied by a system of coalescing random walks evolving on a transitive finite graph. Both examples exhibit a sort of asymptotic lumpability.

start: 2021-09-17T17:30 end: 2021-09-17T18:15 speaker: Johel Beltrán (Pontificia Universidad Católica, Perú) @@ -1797,42 +1797,42 @@ 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. +

The number of roots of random polynomials have been intensively studied for a long time. In the case of systems of polynomial equations the first important results can be traced back to the nineties when Kostlan, Shub and Smale computed the expectation of the number of roots of some random polynomial systems with invariant distributions. Nowadays this is a very active field.

+

In this talk we are concerned with the variance and the asymptotic distribution of the number of roots of invariant polynomial systems. As particular examples we consider Kostlan-Shub-Smale, random spherical harmonics and Real Fubini Study systems.

start: 2021-09-13T16:45-0300 end: 2021-09-13T17:30-0300 speaker: Federico Dalmao Artigas (Universidad de la República, Uruguay) - titulo: Univariate Rational Sum of Squares abstract: | - Landau in 1905 proved that every univariate polynomial with rational coefficients which is strictly positive on the reals is a sum of squares of rational polynomials. However, it is still not known whether univariate rational polynomials which are non-negative on all the reals, rather than strictly positive, are sums of squares of rational polynomials. - In this talk we consider the local counterpart of this problem, namely, we consider rational polynomials that are non-negative on the real roots of another non-zero rational polynomial. Parrilo in 2003 gave a simple construction that implies that if \(f\) in \(\mathbb R[x]\) is squarefree and \(g\) in \(\mathbb R[x]\) is non-negative on the real roots of \(f\) then \(g\) is a sum of squares of real polynomials modulo \(f\). Here, inspired by this construction, we prove that if \(g\) is a univariate rational polynomial which is non-negative on the real roots of a rational polynomial \(f\) (with some condition on \(f\) wrt \(g\) which includes squarefree polynomials) then it is a sum of squares of rational polynomials modulo \(f\). - Joint work with Bernard Mourrain (INRIA, Sophia Antipolis) and Agnes Szanto (North Carolina State University). +

Landau in 1905 proved that every univariate polynomial with rational coefficients which is strictly positive on the reals is a sum of squares of rational polynomials. However, it is still not known whether univariate rational polynomials which are non-negative on all the reals, rather than strictly positive, are sums of squares of rational polynomials.

+

In this talk we consider the local counterpart of this problem, namely, we consider rational polynomials that are non-negative on the real roots of another non-zero rational polynomial. Parrilo in 2003 gave a simple construction that implies that if \(f\) in \(\mathbb R[x]\) is squarefree and \(g\) in \(\mathbb R[x]\) is non-negative on the real roots of \(f\) then \(g\) is a sum of squares of real polynomials modulo \(f\). Here, inspired by this construction, we prove that if \(g\) is a univariate rational polynomial which is non-negative on the real roots of a rational polynomial \(f\) (with some condition on \(f\) wrt \(g\) which includes squarefree polynomials) then it is a sum of squares of rational polynomials modulo \(f\).

+

Joint work with Bernard Mourrain (INRIA, Sophia Antipolis) and Agnes Szanto (North Carolina State University).

start: 2021-09-13T15:45-0300 end: 2021-09-13T16:30-0300 speaker: Teresa Krick (Universidad de Buenos Aires, Argentina) - titulo: On eigenvalues of symmetric matrices with PSD principal submatrices abstract: | - Real symmetric matrices of size n, whose all principal submatrices of size \(k \lt n\) are positive semidefinite, form a closed convex cone. Such matrices do not need to be PSD and, in particular, they can have negative eigenvalues. The geometry of the set of eigenvalues of all such matrices is far from being completely understood. In this talk I will show that already when \((n,k)=(4,2)\) the set of eigenvalues is not convex. +

Real symmetric matrices of size n, whose all principal submatrices of size \(k \lt n\) are positive semidefinite, form a closed convex cone. Such matrices do not need to be PSD and, in particular, they can have negative eigenvalues. The geometry of the set of eigenvalues of all such matrices is far from being completely understood. In this talk I will show that already when \((n,k)=(4,2)\) the set of eigenvalues is not convex.

start: 2021-09-14T15:00-0300 end: 2021-09-14T15:45-0300 speaker: Khazhgali Kozhasov (Technische Universität Braunschweig, Alemania) - titulo: Hausdorff approximation and volume of tubes of singular algebraic sets abstract: | - I will discuss the problem of estimating the volume of the tube around an algebraic set (possibly singular) as a function of the dimension of the set and its degree. In particular I will show bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This problem is related to numerical algebraic geometry, where sets of ill-contidioned inputs are typically described by (possibly singular) algebraic sets, and their neighborhoods describe bad-conditioned inputs. Our result generalizes previous work of Lotz on smooth complete intersections in the euclidean space and of Bürgisser, Cucker and Lotz on hypersurfaces in the sphere, and gives a complete solution to Problem 17 in the book titled "Condition" by Bürgisser and Cucker. This is joint work with S. Basu. +

I will discuss the problem of estimating the volume of the tube around an algebraic set (possibly singular) as a function of the dimension of the set and its degree. In particular I will show bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This problem is related to numerical algebraic geometry, where sets of ill-contidioned inputs are typically described by (possibly singular) algebraic sets, and their neighborhoods describe bad-conditioned inputs. Our result generalizes previous work of Lotz on smooth complete intersections in the euclidean space and of Bürgisser, Cucker and Lotz on hypersurfaces in the sphere, and gives a complete solution to Problem 17 in the book titled "Condition" by Bürgisser and Cucker. This is joint work with S. Basu.

start: 2021-09-13T15:00-0300 end: 2021-09-13T15:45-0300 speaker: Antonio Lerario (Scuola Internazionale Superiore di Studi Avanzati, Italia) - titulo: Algebraic computational problems and conditioning abstract: | - We study the conditioning of algebraic computational problems. We have particular interest in which known theorems about conditioning on particular problems can be extended to the general context. We will pay special attention to the case in which the set of inputs with a common output is not a linear subspace. +

We study the conditioning of algebraic computational problems. We have particular interest in which known theorems about conditioning on particular problems can be extended to the general context. We will pay special attention to the case in which the set of inputs with a common output is not a linear subspace.

start: 2021-09-14T15:45-0300 end: 2021-09-14T16:30-0300 speaker: Federico Carrasco (Universidad de la República, Uruguay) - titulo: Sparse homotopy, toric varieties, and the points at infinity abstract: | - In my previous talk at CLAM-2021, I stated a complexity bound for solving sparse systems of polynomial equations in terms of condition numbers, mixed volume and surface and a few technical details. This results uses a technique that I call renormalization, and the bound excludes systems with solution at toric infinity. - In a recent paper, Duff, Telen, Walker and Yahl proposed a homotopy algorithm using Cox coordinates. This representation associates one new coordinate to each ray (one-dimensional cone) on the fan of the toric variety. - In this talk I will report on recent progress inspired by their work, on non-degenerate solutions at toric infinity for sparse polynomial systems. +

In my previous talk at CLAM-2021, I stated a complexity bound for solving sparse systems of polynomial equations in terms of condition numbers, mixed volume and surface and a few technical details. This results uses a technique that I call renormalization, and the bound excludes systems with solution at toric infinity.

+

In a recent paper, Duff, Telen, Walker and Yahl proposed a homotopy algorithm using Cox coordinates. This representation associates one new coordinate to each ray (one-dimensional cone) on the fan of the toric variety.

+

In this talk I will report on recent progress inspired by their work, on non-degenerate solutions at toric infinity for sparse polynomial systems.

start: 2021-09-14T16:45-0300 end: 2021-09-14T17:30-0300 speaker: Gregorio Malajovich (Universidade Federal do Rio de Janeiro, Brasil) @@ -1944,62 +1944,62 @@ 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. +

Recently, Oguiso addressed the following question, attributed to Gizatullin: ``Which automorphisms of a smooth quartic K3 surface \(D\subset \mathbb{P}^3\) are induced by Cremona transformations of the ambient space \(\mathbb{P}^3\)?'' When \(D\subset \mathbb{P}^3\) is a quartic surface, \((\mathbb{P}^3,D)\) is an example of a \emph{Calabi-Yau pair}, that is, a pair \((X,D)\) consisting of a normal projective variety \(X\) and an effective Weil divisor \(D\) on \(X\) such that \(K_X+D\sim 0\). In this talk, I will explain a general framework to study the birational geometry of mildly singular Calabi-Yau pairs. This is a joint work with Alessio Corti and Alex Massarenti.

start: 2021-09-14T15:00-0300 end: 2021-09-14T15:45-0300 speaker: Carolina Araujo (Instituto de Matemática Pura e Aplicada, Brasil) - titulo: On reconstructing subvarieties from their periods abstract: | - We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. This is a joint work with Emre Sertoz. +

We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. This is a joint work with Emre Sertoz.

start: 2021-09-15T16:45-0300 end: 2021-09-15T17:30-0300 speaker: Hossein Movasati (Instituto de Matemática Pura e Aplicada, Brasil) - titulo: 'Quantum cohomology and derived categories: the case of isotropic Grassmannians' abstract: | - The relation between quantum cohomology and derived categories has been known since long ago. In 1998 Dubrovin formulated his celebrated conjecture relating semisimplicity of the quantum cohomology of Fano manifolds and the existence of exceptional collections in their derived categories of coherent sheaves. In this talk we will discuss recent joint work of the author with Anton Mellit, Nicolas Perrin and Maxim Smirnov studying the relation between quantum cohomology for certain isotropic Grassmannians and their quantum cohomology in the spirit of Dubrovin conjecture. In this case the small quantum cohomology turns out not to be semisimple but still an interesting decomposition of the derived category arises. If time permits new directions and some problems will be mentioned. +

The relation between quantum cohomology and derived categories has been known since long ago. In 1998 Dubrovin formulated his celebrated conjecture relating semisimplicity of the quantum cohomology of Fano manifolds and the existence of exceptional collections in their derived categories of coherent sheaves. In this talk we will discuss recent joint work of the author with Anton Mellit, Nicolas Perrin and Maxim Smirnov studying the relation between quantum cohomology for certain isotropic Grassmannians and their quantum cohomology in the spirit of Dubrovin conjecture. In this case the small quantum cohomology turns out not to be semisimple but still an interesting decomposition of the derived category arises. If time permits new directions and some problems will be mentioned.

start: 2021-09-13T15:45-0300 end: 2021-09-13T16:30-0300 speaker: John Alexander Cruz Morales (Universidad Nacional de Colombia, Colombia) - titulo: Initial degeneration in differential algebraic geometry abstract: | - By a degeneration, we mean a process that transforms a geometric object \(X/F\) defined over a field \(F\) into a simpler object that retains many of the relevant properties of \(X\). Formally, any degeneration is realized by an integral model for \(X\); that is, a flat scheme \(X'/R\) defined over some integral domain \(R\) whose generic fiber is the original object \(X\). - In this talk, we endow the field \(F=K((t_1,\ldots,t_m))\) of quotients of multivariate formal power series with a (generalized) non-Archimedean absolute value \(|\cdot|\). We use this to establish the existence of (affine) integral models \(X'/R\) over the ring of integers \(R=\{|x|\leq 1\}\) for schemes \(X\) associated to solutions of systems of algebraic partial differential equations with coefficients on \(F\). We also concretely describe the specialization map of a model \(X'/R\) to the maximal ideals of \(R\), which are encoded in terms of usual (total) monomial orderings. +

By a degeneration, we mean a process that transforms a geometric object \(X/F\) defined over a field \(F\) into a simpler object that retains many of the relevant properties of \(X\). Formally, any degeneration is realized by an integral model for \(X\); that is, a flat scheme \(X'/R\) defined over some integral domain \(R\) whose generic fiber is the original object \(X\).

+

In this talk, we endow the field \(F=K((t_1,\ldots,t_m))\) of quotients of multivariate formal power series with a (generalized) non-Archimedean absolute value \(|\cdot|\). We use this to establish the existence of (affine) integral models \(X'/R\) over the ring of integers \(R=\{|x|\leq 1\}\) for schemes \(X\) associated to solutions of systems of algebraic partial differential equations with coefficients on \(F\). We also concretely describe the specialization map of a model \(X'/R\) to the maximal ideals of \(R\), which are encoded in terms of usual (total) monomial orderings.

start: 2021-09-13T16:45-0300 end: 2021-09-13T17:30-0300 speaker: Cristhian Garay López (Centro de Investigación en Matemáticas, México) - titulo: Rank two bundles over fibered surfaces abstract: | - Let S be a smooth complex surface fibered over a curve C. Under suitable assumptions, if E is a stable rank two bundle on C then its pullback is a H-stable bundle on S. In this sense we can relate moduli spaces of rank two stable bundles on the surface S with moduli spaces of rank two stable bundles on C. In this talk we aim to take advantage of this relation to provide examples of Brill–Noether loci on fibered surfaces. +

Let S be a smooth complex surface fibered over a curve C. Under suitable assumptions, if E is a stable rank two bundle on C then its pullback is a H-stable bundle on S. In this sense we can relate moduli spaces of rank two stable bundles on the surface S with moduli spaces of rank two stable bundles on C. In this talk we aim to take advantage of this relation to provide examples of Brill–Noether loci on fibered surfaces.

start: 2021-09-14T15:45-0300 end: 2021-09-14T16:30-0300 speaker: Graciela Reyes (Universidad Autónoma de Zacatecas, México) - titulo: Langlands Program and Ramanujan Conjecture abstract: | - We will give an overview of several aspects of the Langlands Program that interconnect different fields of mathematics, namely: Algebraic Geometry, Number Theory and Representation Theory. We study general results over global fields, and mention what is known on Langlands functoriality conjectures. In characteristic p, we have a rich interplay with algebraic geometry and we present results of the author in connection with the work of Laurent and Vincent Lafforgue. As a main application, we look at the Ramanujan conjecture, known for generic representations of the classical groups and GL(n) over function fields, for example. +

We will give an overview of several aspects of the Langlands Program that interconnect different fields of mathematics, namely: Algebraic Geometry, Number Theory and Representation Theory. We study general results over global fields, and mention what is known on Langlands functoriality conjectures. In characteristic p, we have a rich interplay with algebraic geometry and we present results of the author in connection with the work of Laurent and Vincent Lafforgue. As a main application, we look at the Ramanujan conjecture, known for generic representations of the classical groups and GL(n) over function fields, for example.

start: 2021-09-14T16:45-0300 end: 2021-09-14T17:30-0300 speaker: Luis Alberto Lomelí (Pontificia Universidad Católica, Chile) - titulo: Actions and Symmetries abstract: | - I will discuss some classical and some recent results about group and algebra actions on abelian varieties and curves. +

I will discuss some classical and some recent results about group and algebra actions on abelian varieties and curves.

start: 2021-09-13T15:00-0300 end: 2021-09-13T15:45-0300 speaker: Rubí Rodríguez (Universidad de la Frontera, Chile) - titulo: Representation of groups schemes that are affine over an abelian variety abstract: | - The classical representation theory of affine schemes over a field, has been widely considered and many of its main properties have been understood and put under contral--after the work of many mathematicians along the second half of the 20th century. In joint work with del Ángel and Rittatore, we proposed a representation theory for the more general situation where the group scheme is affine over a fixed abelian variety A (the classical case corresponds to the situation that A is the trivial abelian variety with only one point). In this talk we describe the main relevant definitions, where the representations are vector bundles over A with suitable actions of the group scheme G. Then, we show that in our context an adequate formulation of Tannaka--type of recognition and reconstruction results (due to Grothendieck, Saavedra, Deligne and others) remain valid. +

The classical representation theory of affine schemes over a field, has been widely considered and many of its main properties have been understood and put under contral--after the work of many mathematicians along the second half of the 20th century. In joint work with del Ángel and Rittatore, we proposed a representation theory for the more general situation where the group scheme is affine over a fixed abelian variety A (the classical case corresponds to the situation that A is the trivial abelian variety with only one point). In this talk we describe the main relevant definitions, where the representations are vector bundles over A with suitable actions of the group scheme G. Then, we show that in our context an adequate formulation of Tannaka--type of recognition and reconstruction results (due to Grothendieck, Saavedra, Deligne and others) remain valid.

start: 2021-09-15T15:45-0300 end: 2021-09-15T16:30-0300 speaker: Walter Ferrer (Universidad de la República, Uruguay) - titulo: A criterion for an abelian variety to be non-simple and applications abstract: | - Let \((A,\cL)\) 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. +

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: 2021-09-15T15:00-0300 end: 2021-09-15T15:45-0300 speaker: Anita Rojas (Universidad de Chile, Chile) @@ -2012,55 +2012,55 @@ 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. +

El formato general de los problemas de clasificación que nos ocupan es el siguiente. Se tienen un anillo de base \(\ell\), una familia \(C\) de \(\ell\)-álgebras, y un invariante \(K\)-teórico (e.g. \(K_0\) con su preorden natural), y nos preguntamos si dos integrantes de la familia con invariantes isomorfos son necesariamente isomorfas o equivalentes bajo algún criterio (e.g. Morita equivalentes). Una fuente de estos problemas es tomar una clase de \(C^*\)-álgebras (e.g. completaciones de \(\mathbb{C}\)-álgebras definidas por cierto tipo de generadores y relaciones) que ha sido clasificada por su \(K\)-teoria (topológica) y preguntarse si la \(K\)-teoría algebraica clasifica el análogo puramente algebraico de esa familia (e.g. la dada por el mismo tipo de generadores y relaciones, sin completar). En la charla veremos algunos ejemplos de problemas de ese tipo y algunos resultados.

start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Guillermo Cortiñas (Universidad de Buenos Aires, Argentina) - titulo: Algebraic K theory of 3-manifold groups abstract: | - We describe the algebraic K-theory groups \(K_i(Z[G[))\) where G is the fundamental group of a closed compact 3-manifold and all \(i\in Z\). +

We describe the algebraic K-theory groups \(K_i(Z[G[))\) where G is the fundamental group of a closed compact 3-manifold and all \(i\in Z\).

start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Daniel Juan (Universidad Nacional Autónoma de México, México) - titulo: Clasificación homotópica de álgebras de Leavitt simples puramente infinitas abstract: | - En esta charla hablaremos del problema de clasificación de álgebras de Leavitt simples puramente infinitas sobre un grafo finito \(E\) y un cuerpo \(\ell\). Dado un grafo \(E\) tenemos asociado una \(\ell\)-álgebra de Leavitt \(L(E)\). Es un problema abierto decidir cuando el par \((K_0(E), [1_{L(E)}])\), que consiste del grupo de Grothendieck junto con la clase \([1_{L(E)}]\) de la identidad es un invariante completo, a menos de isomorfismo, para la clasificación de álgebras de Leavitt simples puramente infintas de grafo finito. Mostraremos que el par \((K_0(E), [1_{L(E)}])\) es un invariante completo para la clasificación a menos de equivalencia homotópica polinomial. Para probar esto vamos a estudiar la \(K\)-teoría álgebraica bivariante entre álgebras de Leavitt. +

En esta charla hablaremos del problema de clasificación de álgebras de Leavitt simples puramente infinitas sobre un grafo finito \(E\) y un cuerpo \(\ell\). Dado un grafo \(E\) tenemos asociado una \(\ell\)-álgebra de Leavitt \(L(E)\). Es un problema abierto decidir cuando el par \((K_0(E), [1_{L(E)}])\), que consiste del grupo de Grothendieck junto con la clase \([1_{L(E)}]\) de la identidad es un invariante completo, a menos de isomorfismo, para la clasificación de álgebras de Leavitt simples puramente infintas de grafo finito. Mostraremos que el par \((K_0(E), [1_{L(E)}])\) es un invariante completo para la clasificación a menos de equivalencia homotópica polinomial. Para probar esto vamos a estudiar la \(K\)-teoría álgebraica bivariante entre álgebras de Leavitt.

start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Diego Montero (Despegar, Argentina) - titulo: The algebraic K-theory of the Hilbert modular group abstract: | - In this talk I will describe how to compute the algebraic K-theory of the Hilbert modular group using the Farrell-Jones conjecture. Among the tools we use are the p-chain spectral sequence, models for the classifying space for the family of virtually cyclic subgroups, and the inductive structure of the equivariant homology theory that appears in the statement of the Farrell-Jones conjecture. This is joint work with M. Bustamante and M. Velásquez +

In this talk I will describe how to compute the algebraic K-theory of the Hilbert modular group using the Farrell-Jones conjecture. Among the tools we use are the p-chain spectral sequence, models for the classifying space for the family of virtually cyclic subgroups, and the inductive structure of the equivariant homology theory that appears in the statement of the Farrell-Jones conjecture. This is joint work with M. Bustamante and M. Velásquez

start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Luis Jorge Sanchez Saldaña (Universidad Nacional Autónoma de México, México) - titulo: A Gillet-Waldhausen Theorem for chain complexes of sets abstract: | - In recent work, Campbell and Zakharevich introduced a new type of structure, called ACGW-category. These are double categories satisfying a list of axioms that seek to extract the properties of abelian categories which make them so particularly well-suited for algebraic K-theory. The main appeal of these double categories is that they generalize the structure of exact sequences in abelian categories to non-additive settings such as finite sets and reduced schemes, thus showing how finite sets and schemes behave like the objects of an exact category for the purpose of algebraic K-theory. - In this talk, we will explore the key features of ACGW categories and the intuition behind them. Then, we will move on to ongoing work with Brandon Shapiro where we further develop this program by defining chain complexes and quasi-isomorphisms for finite sets. These satisfy an analogue of the classical Gillet--Waldhausen Theorem, providing an alternate model for the K-theory of finite sets. +

In recent work, Campbell and Zakharevich introduced a new type of structure, called ACGW-category. These are double categories satisfying a list of axioms that seek to extract the properties of abelian categories which make them so particularly well-suited for algebraic K-theory. The main appeal of these double categories is that they generalize the structure of exact sequences in abelian categories to non-additive settings such as finite sets and reduced schemes, thus showing how finite sets and schemes behave like the objects of an exact category for the purpose of algebraic K-theory.

+

In this talk, we will explore the key features of ACGW categories and the intuition behind them. Then, we will move on to ongoing work with Brandon Shapiro where we further develop this program by defining chain complexes and quasi-isomorphisms for finite sets. These satisfy an analogue of the classical Gillet--Waldhausen Theorem, providing an alternate model for the K-theory of finite sets.

start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Maru Sarazola (Johns Hopkins University, Estados Unidos) - titulo: Homología de Hochschild topológica y métodos de traza abstract: | - Para un anillo \(R\), o más en general un espectro en anillos ("ring spectrum"), Quillen construyó la K-teoría algebraica superior de \(R\) como un espectro en anillos \(K(R)\). Se puede construir otro espectro \(THH(R)\), la homología de Hochschild topológica de \(R\): es un análogo de la homología de Hochschild clásica, solo que en vez de ser sobre el anillo de los enteros \(\mathbb{Z}\), es sobre una base más profunda, el espectro en anillos \(S\), llamado "espectro de las esferas" ("sphere spectrum"). La relación está dada por un mapa \(K(R) \rightarrow THH(R)\) llamado traza, o traza de Dennis topológica, que a veces permite acercarse al cálculo de \(K(R)\). Esbozaremos estas construcciones y daremos un ejemplo de cálculo. +

Para un anillo \(R\), o más en general un espectro en anillos ("ring spectrum"), Quillen construyó la K-teoría algebraica superior de \(R\) como un espectro en anillos \(K(R)\). Se puede construir otro espectro \(THH(R)\), la homología de Hochschild topológica de \(R\): es un análogo de la homología de Hochschild clásica, solo que en vez de ser sobre el anillo de los enteros \(\mathbb{Z}\), es sobre una base más profunda, el espectro en anillos \(S\), llamado "espectro de las esferas" ("sphere spectrum"). La relación está dada por un mapa \(K(R) \rightarrow THH(R)\) llamado traza, o traza de Dennis topológica, que a veces permite acercarse al cálculo de \(K(R)\). Esbozaremos estas construcciones y daremos un ejemplo de cálculo.

start: 2021-09-14T15:00 end: 2021-09-14T15:45 speaker: Bruno Stonek (Instituto de Matemáticas de la Academia Polaca de Ciencias, Polonia) - titulo: Topología controlada y conjeturas de isomorfismo en \(K\)-teoría abstract: | - Sean \(R\) un anillo, \(G\) un grupo, \(\mathcal{F}\) una familia de subgrupos y \(E_{\mathcal{F}}G\) el \(G\)-\(CW\)-complejo universal con isotropía en \(\mathcal{F}\). La conjetura de isomorfismo identifica, mediante un morfismo de ensamble, la \(K\)-teoría del anillo de grupo \(\mathbf{K}(RG)\) con una teoría de homología equivariante evaluada en \(E_{\mathcal{F}}G\). Una construcción de esta teoría de homología se puede hacer utilizando topología controlada. En este contexto la conjetura afirma que el morfismo de ensamble +

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 + 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.) + que resulta un isomorfismo para R regular. (Trabajo en conjunto con E. Ellis, E. Rodríguez Cirone y S. Vega.)

start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Gisela Tartaglia (Universidad Nacional de La Plata, Argentina) - titulo: Proper actions and equivariant K-theory of group extensions abstract: | - In this talk we present some decompositions appearing in equivariant K-theory. Let \[0\to A\to G\to Q\to0\] be an extension of Lie groups with \(A\) compact, let \(X\) be proper \(Q\)-space, we present a decomposition of the \(G\)-equivariant K-theory of \(X\) in terms of twisted equivariant K-theories of \(X\) respect to certain subgroups of \(Q\). We present some examples of computations using this decomposition. Finally we give some ideas of how to extend this decomposition in the context of C*-algebras. +

In this talk we present some decompositions appearing in equivariant K-theory. Let \[0\to A\to G\to Q\to0\] be an extension of Lie groups with \(A\) compact, let \(X\) be proper \(Q\)-space, we present a decomposition of the \(G\)-equivariant K-theory of \(X\) in terms of twisted equivariant K-theories of \(X\) respect to certain subgroups of \(Q\). We present some examples of computations using this decomposition. Finally we give some ideas of how to extend this decomposition in the context of C*-algebras.

start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Mario Velazquez (Universität Göttingen, Alemania) @@ -2228,52 +2228,52 @@ 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. +

A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) or pluriclosed if the torsion of the associated Bismut connection associated is closed. I will present some general results on pluriclosed metrics in relation to symplectic geometry, the pluriclosed flow and Kähler-like curvature conditions.

start: 2021-09-13T15:00 end: 2021-09-13T15:45 speaker: Anna Fino (Università di Torino, Italia) - titulo: The prescribed cross curvature problem abstract: | - Chow and Hamilton introduced the notion of cross curvature and the associated geometric flow in 2004. Several authors have built on their work to study the uniformisation of negatively curved manifolds, Dehn fillings, and other topics. Hamilton conjectured that it is always possible to find a metric with given positive cross curvature on the three-sphere and that such a metric is unique. We will discuss several results that support the existence portion of this conjecture. Next, we will produce a counterexample showing that uniqueness fails in general. +

Chow and Hamilton introduced the notion of cross curvature and the associated geometric flow in 2004. Several authors have built on their work to study the uniformisation of negatively curved manifolds, Dehn fillings, and other topics. Hamilton conjectured that it is always possible to find a metric with given positive cross curvature on the three-sphere and that such a metric is unique. We will discuss several results that support the existence portion of this conjecture. Next, we will produce a counterexample showing that uniqueness fails in general.

start: 2021-09-14T17:30 end: 2021-09-14T18:15 speaker: Artem Pulemotov (The University of Queensland, Australia), joint with Timothy Buttsworth (The University of Queensland) - titulo: Soliton solutions to the curve shortening flow on the 2-dimensional hyperbolic space abstract: | - We prove that a curve is a soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space if and only if its geodesic curvature is given as the inner product between its tangent vector field and a vector of the 3-dimensional Minkowski space. We prove that there are three classes of such solutions and for each fixed vector there exits a 2-parameter family of soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space. Moreover, we prove that each soliton is defined on the whole real line, it is embedded and its geodesic curvature, at each end, converges to a constant +

We prove that a curve is a soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space if and only if its geodesic curvature is given as the inner product between its tangent vector field and a vector of the 3-dimensional Minkowski space. We prove that there are three classes of such solutions and for each fixed vector there exits a 2-parameter family of soliton solution to the curve shortening flow on the 2-dimensional hyperbolic space. Moreover, we prove that each soliton is defined on the whole real line, it is embedded and its geodesic curvature, at each end, converges to a constant

start: 2021-09-14T15:00 end: 2021-09-13T15:45 speaker: Keti Tenenblat (Universidade de Brasília, Brasil), joint with Fabio Nunes da Silva (Universidade Federal do Oeste da Bahia) - titulo: Upper bound on the revised first Betti number and torus stability for RCD spaces abstract: | - Gromov and Gallot showed that for a fixed dimension \(n\) there exists a number \(\varepsilon(n)>0\) so that any \(n\)-dimensional riemannian manifold \((M,g)\) satisfying \(\textrm{Ric}_g \textrm{diam}(M,g)^2 \geq -\varepsilon(n)\) has first Betti number smaller than or equal to \(n\). In the equality case, \(\textrm{b}_1(M)=n\), Cheeger and Colding showed that then \(M\) has to be bi-Holder homeomorphic to a flat torus. This part can be seen as a stability statement to the rigidity result proven by Bochner, namely, closed riemannian manifolds with nonnegative Ricci curvature and first Betti number equal to their dimension have to be a torus. - The proofs of Gromov and, Cheeger and Colding rely on finding an appropriate subgroup of the abelianized fundamental group to pass to a nice covering space of \(M\) and then study the geometry of the covering. In this talk we will generalize these results to the case of \(RCD(K,N)\) spaces, which is the synthetic notion of riemannian manifolds satisfying \(\text{Ric} \geq K\) and \(\text{dim} \leq N\). This class of spaces include Ricci limit spaces and Alexandrov spaces. +

Gromov and Gallot showed that for a fixed dimension \(n\) there exists a number \(\varepsilon(n)>0\) so that any \(n\)-dimensional riemannian manifold \((M,g)\) satisfying \(\textrm{Ric}_g \textrm{diam}(M,g)^2 \geq -\varepsilon(n)\) has first Betti number smaller than or equal to \(n\). In the equality case, \(\textrm{b}_1(M)=n\), Cheeger and Colding showed that then \(M\) has to be bi-Holder homeomorphic to a flat torus. This part can be seen as a stability statement to the rigidity result proven by Bochner, namely, closed riemannian manifolds with nonnegative Ricci curvature and first Betti number equal to their dimension have to be a torus.

+

The proofs of Gromov and, Cheeger and Colding rely on finding an appropriate subgroup of the abelianized fundamental group to pass to a nice covering space of \(M\) and then study the geometry of the covering. In this talk we will generalize these results to the case of \(RCD(K,N)\) spaces, which is the synthetic notion of riemannian manifolds satisfying \(\text{Ric} \geq K\) and \(\text{dim} \leq N\). This class of spaces include Ricci limit spaces and Alexandrov spaces.

start: 2021-09-14T16:45 end: 2021-09-14T17:30 speaker: Raquel Perales (Universidad Nacional Autónoma de México, México), joint with Ilaria Mondello (Université de Paris Est Créteil) and Andrea Mondino (Oxford University) - titulo: Left-invariant metrics on six-dimensional nilpotent Lie groups abstract: | - In this talk we determine the moduli space, up to isometric automorphism, of left-invariant metrics on a family of \(6\)-dimensional Lie group \(H\). We also investigate which of these metrics are Hermitian and classify the corresponding complex structures. This talk is based on a joint work with Silvio Reggiani, based on ``The moduli space of left-invariant metrics of a class of six-dimensional nilpotent Lie groups'', arXiv:2011.02854 +

In this talk we determine the moduli space, up to isometric automorphism, of left-invariant metrics on a family of \(6\)-dimensional Lie group \(H\). We also investigate which of these metrics are Hermitian and classify the corresponding complex structures. This talk is based on a joint work with Silvio Reggiani, based on ``The moduli space of left-invariant metrics of a class of six-dimensional nilpotent Lie groups'', arXiv:2011.02854

start: 2021-09-14T15:45 end: 2021-09-14T16:30 speaker: Francisco Vittone (Universidad Nacional de Rosario, Argentina), joint with Silvio Reggiani (Universidad Nacional de Rosario) - titulo: On the structure of homogeneous Riemannian manifolds with nullity abstract: | - We will speak about a joint project, with {\it Antonio J. Di Scala} and {\it Francisco Vittone}, for the study of the structure of irreducible homogeneous Riemannian manifolds \(M^n= G/H\) whose curvature tensor has a non-trivial nullity. In a recent paper we developed a general theory to deal with such spaces. By making use of this theory we were able to construct the first non-trivial examples in any dimension. The key fact is the existence of a non-trivial transvection \(X\) at \(p\) (i.e.\((\nabla X)_p = 0\)) such that \(X_p\notin \nu_p\) (the nullity subspace at \(p\)), but the Jacobi operator of \(X_p\) is zero. The nullity distribution \(\nu\) is highly non-homogeneus in the sense that no non-trivial Killing field lie in \(\nu\) (an so \(\nu\) is not given by the orbits of an isometry subgroup of \(G\)). One has that the Lie algebra \(\mathfrak g\) of \(G\) is never reductive. The co-nullity index \(k\) must be always at least \(3\) and if \(k=3\), then \(G\) must be solvable and \(H\) trivial. The leaves of the nullity foliation \(\mathcal F\) are closed and isometric to a Euclidean space. Moreover, the stabilizer of a given leaf acts effectively on the quotient space \(M/\mathcal F\) and so it does not admit a Riemannian \(G\)-invariant metric. Our main new result is that the so-called {\it adapted} tranvections lie in an abelian ideal \(\mathfrak a\) of \(\mathfrak g\). The distribution \(q\mapsto \mathfrak a .q + \nu _q\) is integrable and does not depend on the presentation group \(G\) and so it defines a geometric foliation \(\hat {\mathcal F}\) on \(M\) (by taking the clausure of the leaves). Moreover, the nullity \(\nu\) is parallel along any leaf of \(\hat {\mathcal F}\) and the projection to the quotient space \(M/ \hat {\mathcal F}\) is a Riemannian submersion. We intend to relate the geometry of \(M\) to that of this quotient. +

We will speak about a joint project, with {\it Antonio J. Di Scala} and {\it Francisco Vittone}, for the study of the structure of irreducible homogeneous Riemannian manifolds \(M^n= G/H\) whose curvature tensor has a non-trivial nullity. In a recent paper we developed a general theory to deal with such spaces. By making use of this theory we were able to construct the first non-trivial examples in any dimension. The key fact is the existence of a non-trivial transvection \(X\) at \(p\) (i.e.\((\nabla X)_p = 0\)) such that \(X_p\notin \nu_p\) (the nullity subspace at \(p\)), but the Jacobi operator of \(X_p\) is zero. The nullity distribution \(\nu\) is highly non-homogeneus in the sense that no non-trivial Killing field lie in \(\nu\) (an so \(\nu\) is not given by the orbits of an isometry subgroup of \(G\)). One has that the Lie algebra \(\mathfrak g\) of \(G\) is never reductive. The co-nullity index \(k\) must be always at least \(3\) and if \(k=3\), then \(G\) must be solvable and \(H\) trivial. The leaves of the nullity foliation \(\mathcal F\) are closed and isometric to a Euclidean space. Moreover, the stabilizer of a given leaf acts effectively on the quotient space \(M/\mathcal F\) and so it does not admit a Riemannian \(G\)-invariant metric. Our main new result is that the so-called {\it adapted} tranvections lie in an abelian ideal \(\mathfrak a\) of \(\mathfrak g\). The distribution \(q\mapsto \mathfrak a .q + \nu _q\) is integrable and does not depend on the presentation group \(G\) and so it defines a geometric foliation \(\hat {\mathcal F}\) on \(M\) (by taking the clausure of the leaves). Moreover, the nullity \(\nu\) is parallel along any leaf of \(\hat {\mathcal F}\) and the projection to the quotient space \(M/ \hat {\mathcal F}\) is a Riemannian submersion. We intend to relate the geometry of \(M\) to that of this quotient.

start: 2021-09-13T17:30 end: 2021-09-13T18:15 speaker: Carlos E. Olmos (Universidad Nacional de Córdoba, Argentina) - titulo: Hermitian and Complex Geometry in Flag Manifolds abstract: | - The subject of this talk is a series of results on Hermitian, Complex and Symplectic geometries on the flag manifolds of noncompact semi-simple Lie groups (real or complex). Equations of Differential Geometry in coset spaces of Lie groups are reduced to algebraic equations by homogeneity. Thus many questions are stated and solved in the realm of algebraic properties of the isotropy representation at the tangent space at the origin. In case of the flag manifolds these representations are usually dealt with a combinatorics involving the algebraic structure of the root systems and their Weyl groups. - In this talk some questions and results in this direction will be worked out. Best results are obtained for flag manifolds of the complex groups because the isotropy representations are better behaved. Results on real flag manifolds are more sparse and technically harder. In the complex case it will be presented a more detailed account on the \((1,2)\)-symplectic Hermitian metrics on the maximal (full) flag manifolds. +

The subject of this talk is a series of results on Hermitian, Complex and Symplectic geometries on the flag manifolds of noncompact semi-simple Lie groups (real or complex). Equations of Differential Geometry in coset spaces of Lie groups are reduced to algebraic equations by homogeneity. Thus many questions are stated and solved in the realm of algebraic properties of the isotropy representation at the tangent space at the origin. In case of the flag manifolds these representations are usually dealt with a combinatorics involving the algebraic structure of the root systems and their Weyl groups.

+

In this talk some questions and results in this direction will be worked out. Best results are obtained for flag manifolds of the complex groups because the isotropy representations are better behaved. Results on real flag manifolds are more sparse and technically harder. In the complex case it will be presented a more detailed account on the \((1,2)\)-symplectic Hermitian metrics on the maximal (full) flag manifolds.

start: 2021-09-13T15:45 end: 2021-09-13T16:30 speaker: Luiz San Martin (Universidade Estadual de Campinas, Brasil) - titulo: Stability of Einstein metrics on flag manifolds with \(b_2(M)=1\) abstract: | - Let \(M\) be a compact differentiable manifold and let \(\mathcal{M}\) denote the space of all unit volume Riemannian metrics on \(M\). Back in 1915, Hilbert proved that the critical points of the simplest curvature functional, given by the total scalar curvature \(Sc:\mathcal{M}_1\rightarrow {\Bbb R}\), are precisely Einstein metrics. - In this talk, after some general preliminaries, we will focus on the case when the metrics and the variations are considered to be \(G\)-invariant for some compact Lie group \(G\) acting transitively on \(M\). As an application, we will give the stability and critical point types of all Einstein metrics on flag manifolds with \(b_2(M)=1\). +

Let \(M\) be a compact differentiable manifold and let \(\mathcal{M}\) denote the space of all unit volume Riemannian metrics on \(M\). Back in 1915, Hilbert proved that the critical points of the simplest curvature functional, given by the total scalar curvature \(Sc:\mathcal{M}_1\rightarrow {\Bbb R}\), are precisely Einstein metrics.

+

In this talk, after some general preliminaries, we will focus on the case when the metrics and the variations are considered to be \(G\)-invariant for some compact Lie group \(G\) acting transitively on \(M\). As an application, we will give the stability and critical point types of all Einstein metrics on flag manifolds with \(b_2(M)=1\).

start: 2021-09-13T16:45 end: 2021-09-13T17:30 speaker: Jorge Lauret (Universidad Nacional de Córdoba, Argentina) diff --git a/public/style/style.css b/public/style/style.css index 82fbacf..15b0bfc 100644 --- a/public/style/style.css +++ b/public/style/style.css @@ -576,7 +576,7 @@ p.abstract{ max-width:48em; white-space:pre-wrap; } -.abstract > p {font-size:14px;} +.abstract > p {font-size:14px;padding-bottom: 10px;} .avisomodal > h4.aviso-title { text-decoration:underline; color:black;