Seminario de Análisis y Geometría
Los seminarios de Análisis y Geometría se llevan a cabo los días jueves a las 16:10 en la Sala 2 de la Facultad de Matemáticas, Pontificia Universidad Católica de Chile.Organizadores: Pedro Gaspar y Gianmarco Sperone
2023-05-30
17:00hrs.
17:00hrs.
Julio Rossi. Departamento de Matemática, Universidad de Buenos Aires
Convexity and Partial Differential Equations
Sala 1
Abstract:
In this talk we will introduce different notions of convexity that interpolate between classical convexity and quasiconvexity and that, moreover, have a natural fractional extension. For these notions of convexity we also characterize the convex envelope inside a domain of a boundary datum in terms of being the unique viscosity solution to an associated differential equation.
Convexity and Partial Differential Equations
Sala 1
Abstract:
In this talk we will introduce different notions of convexity that interpolate between classical convexity and quasiconvexity and that, moreover, have a natural fractional extension. For these notions of convexity we also characterize the convex envelope inside a domain of a boundary datum in terms of being the unique viscosity solution to an associated differential equation.
2023-05-23
16:00hrs.
16:00hrs.
Duvan Henao. Instituto de Ciencias de la Ingeniería, Universidad de O'higgins
Dipolos armónicos en elasticidad
Sala 1
Abstract:
Whenever the stored energy density of a hyperelastic material has slow growth at infinity (below |F|^p with p less than the space dimension), it may undergo cavitation (the nucleation and sudden growth of internal voids) under large hydrostatic tension [Ball, 1982; James & Spector, 1992]. This constitutes a failure of quasiconvexity and, hence, a challenge for the existence theory in elastostatics [Ball & Murat, 1984]. The obstacle has been overcome under certain coercivity hypotheses [Müller & Spector, 1995; Sivaloganathan & Spector, 2000] which, however, fail to be satisfied by the paradigmatic example in elasticity: that of 3D neo-Hookean materials. A joint work with Marco Barchiesi, Carlos Mora-Corral, and Rémy Rodiac will be presented, where this borderline case was solved for hollow axisymmetric domains. Partial results leading to a solution when the axis of rotation is contained (where the dipoles found by [Conti & De Lellis, 2003] must be proved to be non energy-minimizing) will also be discussed.
Dipolos armónicos en elasticidad
Sala 1
Abstract:
Whenever the stored energy density of a hyperelastic material has slow growth at infinity (below |F|^p with p less than the space dimension), it may undergo cavitation (the nucleation and sudden growth of internal voids) under large hydrostatic tension [Ball, 1982; James & Spector, 1992]. This constitutes a failure of quasiconvexity and, hence, a challenge for the existence theory in elastostatics [Ball & Murat, 1984]. The obstacle has been overcome under certain coercivity hypotheses [Müller & Spector, 1995; Sivaloganathan & Spector, 2000] which, however, fail to be satisfied by the paradigmatic example in elasticity: that of 3D neo-Hookean materials. A joint work with Marco Barchiesi, Carlos Mora-Corral, and Rémy Rodiac will be presented, where this borderline case was solved for hollow axisymmetric domains. Partial results leading to a solution when the axis of rotation is contained (where the dipoles found by [Conti & De Lellis, 2003] must be proved to be non energy-minimizing) will also be discussed.
2023-05-16
16:00hrs.
16:00hrs.
Rafael Benguria. Instituto de Física, PUC de Chile
A variational formulation for Dirac Operators in bounded domains and applications to spectral geometric inequalities
Sala 1
Abstract:
A variational formulation for Dirac Operators in bounded domains and applications to spectral geometric inequalities
Sala 1
Abstract:
In this talk I will present spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. A non-linear variational formulation to characterize its principal eigenvalue will be presented. This characterization allows for a simple proof of a Szegö type inequality as well
as a new formulation of a Faber-Krahn type inequality for this operator. Moreover, strong numerical evidence supporting the existence of a Faber-Krahn type inequality, will be given.
This talk is based on joint work with Pedro Antunes, Vladimir Lotoreichik, and Thomas Ourmières-Bonafos.
2023-05-09
16:00hrs.
16:00hrs.
Pedro Gaspar. Facultad de Matemáticas, PUC de Chile
Connecting unstable minimal surfaces via phase transitions
Sala 1
Abstract:
The Allen–Cahn equation is a semilinear evolution partial differential equation that models phase transition phenomena and which provides a regularization for the mean curvature flow (MCF), one of the most studied geometric flows. In this talk, inspired by Morse-theoretical considerations, we will study eternal solutions to the Allen–Cahn equation connecting unstable equilibria. We will also discuss how to use these flows to construct a family of (weak) eternal solutions to the MCF connecting minimal surfaces of low area in the 3-sphere. This is joint work with Jingwen Chen.
Connecting unstable minimal surfaces via phase transitions
Sala 1
Abstract:
The Allen–Cahn equation is a semilinear evolution partial differential equation that models phase transition phenomena and which provides a regularization for the mean curvature flow (MCF), one of the most studied geometric flows. In this talk, inspired by Morse-theoretical considerations, we will study eternal solutions to the Allen–Cahn equation connecting unstable equilibria. We will also discuss how to use these flows to construct a family of (weak) eternal solutions to the MCF connecting minimal surfaces of low area in the 3-sphere. This is joint work with Jingwen Chen.
2023-04-18
16:00hrs.
16:00hrs.
Raquel Perales. Conacyt-Imate Unam, México
Distancia intrínseca plana y algunas aplicaciones
https://zoom.us/j/95825353771?pwd=VEJBYytjTVZFZFhCSzFwS3d3UVo5Zz09
Abstract:
En esta plática vamos a definir a la distancia intrínseca plana entre dos variedades compactas orientadas que fue formulada por Christina Sormani y Stefan Wenger, y mencionaremos dos resultados de estabilidad que se obtienen utilizando esta distancia.
Distancia intrínseca plana y algunas aplicaciones
https://zoom.us/j/95825353771?pwd=VEJBYytjTVZFZFhCSzFwS3d3UVo5Zz09
Abstract:
En esta plática vamos a definir a la distancia intrínseca plana entre dos variedades compactas orientadas que fue formulada por Christina Sormani y Stefan Wenger, y mencionaremos dos resultados de estabilidad que se obtienen utilizando esta distancia.
2023-03-21
16:00hrs.
16:00hrs.
Paul Pegon. Paris Dauphine
Convergence rate of general entropy-regularized optimal transport costs
Sala 2, edificio Rolando Chuaqui
Abstract:
The entropic regularization of optimal transport is widely popular to recover approximate solutions of optimal transport problems, due to the existence of simple numerical schemes based on Sinkhorn's algorithm. In this talk, I will focus on the the asymptotics of the optimal transport cost as the noise parameter $\varepsilon$ vanishes. The asymptotics is known up to order 2 for cost functions that are sufficiently regular and twisted, such as the quadratic cost. I will present upper and lower bounds for classes of cost functions that are less regular (Lipschitz, with Lipschitz gradients, or non-degenerate), which will depend on the dimension of the marginals but not on the optimal transport plans themselves. Such estimates are in particular relevant to debias regularized optimal transport using Sinkhorn divergences. This is a joint work with G.Carlier and L.Tamanini.
Convergence rate of general entropy-regularized optimal transport costs
Sala 2, edificio Rolando Chuaqui
Abstract:
The entropic regularization of optimal transport is widely popular to recover approximate solutions of optimal transport problems, due to the existence of simple numerical schemes based on Sinkhorn's algorithm. In this talk, I will focus on the the asymptotics of the optimal transport cost as the noise parameter $\varepsilon$ vanishes. The asymptotics is known up to order 2 for cost functions that are sufficiently regular and twisted, such as the quadratic cost. I will present upper and lower bounds for classes of cost functions that are less regular (Lipschitz, with Lipschitz gradients, or non-degenerate), which will depend on the dimension of the marginals but not on the optimal transport plans themselves. Such estimates are in particular relevant to debias regularized optimal transport using Sinkhorn divergences. This is a joint work with G.Carlier and L.Tamanini.
2022-12-06
16:00hrs.
16:00hrs.
Pilar Herreros. PUC Santiago
Multiplicidad de soluciones por cambios de magnitud
Sala 2, edificio Rolando Chuaqui
Abstract:
Multiplicidad de soluciones por cambios de magnitud
Sala 2, edificio Rolando Chuaqui
Abstract:
Estudiaremos las soluciones radialmente simétricas del problema
Δu+f(u)=0, x∈R^N, N>2, lim|x|→∞ u(x)=0.
Veremos que podemos generar nuevas soluciones del problema si introducimos cambios bruscos en la magnitud de la función f. Usando esto construiremos funciones f, definidas por partes, tales que el problema tiene cualquier número pre-determinado de soluciones.
Δu+f(u)=0, x∈R^N, N>2, lim|x|→∞ u(x)=0.
Veremos que podemos generar nuevas soluciones del problema si introducimos cambios bruscos en la magnitud de la función f. Usando esto construiremos funciones f, definidas por partes, tales que el problema tiene cualquier número pre-determinado de soluciones.
2022-11-22
16:00hrs.
16:00hrs.
Arturo Benson. Universidad de Santiago de Chile
Inmersiones de Superficies en Espacios de Curvartura Seccional Constante y su Modelación: Ecuaciones de Tipo Pseudo-esférico y Ecuaciones que Describen Superficies en S^3
Sala 2, edificio Rolando Chuaqui
Abstract:
Inmersiones de Superficies en Espacios de Curvartura Seccional Constante y su Modelación: Ecuaciones de Tipo Pseudo-esférico y Ecuaciones que Describen Superficies en S^3
Sala 2, edificio Rolando Chuaqui
Abstract:
En esta charla presentaré un estudio de inmersiones de superficies pseudo esféricas en $R^3$ y una clasificación de ecuaciones de tipo Lund-Regge que describen
superficies inmersas en la tres-esfera $S^3$. Veremos cómo modelar mediante técnicas de geometría discreta algunas ecuaciones pseudo esféricas como la ecuación de Sine-Gordon y la ecuación de Burgers.
2022-11-15
16:00hrs.
16:00hrs.
Benjamín Palacios. PUC Santiago
Estudio de la tomografía foto-acústica a través del análisis microlocal
Sala 2, edificio Rolando Chuaqui
Abstract:
Estudio de la tomografía foto-acústica a través del análisis microlocal
Sala 2, edificio Rolando Chuaqui
Abstract:
La tomografía foto-acústica (y también la tomografía termo-acústica) es una modalidad de imagen médica en desarrollo que se basa en el acoplamiento ondas electromagnéticas y acústicas por medio del efecto foto-acústico. En términos matemáticos, una parte de esta técnica se modela como un problema inverso de recuperación de una fuente inicial de ultrasonido, donde se asume que el medio de propagación es heterogéneo y las observaciones son adquiridas en una frontera, parcial o totalmente conteniendo el cuerpo en estudio.
En esta charla explicaré cómo el análisis microlocal nos ayuda a entender completamente el problema inverso en su formulación matemática original, y cómo esta metodología de adapta al caso de medios acústicamente atenuantes.
2022-10-24
11:30hrs.
11:30hrs.
Mauricio Godoy. Universidad de la Frontera
Grafos dirigidos y álgebras de Lie de paso 2
Sala 1
Abstract:
En particular, la álgebra de derivaciones de un álgebra de Lie nilpotente gradada que preservan la gradación es de gran utilidad en el estudio de simetrías infinitesimales de sistemas diferenciales, en base a la teoría de prolongaciones de Tanaka.
Grafos dirigidos y álgebras de Lie de paso 2
Sala 1
Abstract:
Las álgebras de Lie nilpotentes son objetos de gran interés en geometría diferencial y análisis geométrico. En este sentido, es importante tener diferentes formas de generar ejemplos importantes, y entender posibles interrelaciones con otras áreas de la matemática, puesto que problemas que aparecen en contextos diferentes pueden encontrar soluciones inesperadas al mirar un mismo objeto desde otras perspectivas.
En particular, la álgebra de derivaciones de un álgebra de Lie nilpotente gradada que preservan la gradación es de gran utilidad en el estudio de simetrías infinitesimales de sistemas diferenciales, en base a la teoría de prolongaciones de Tanaka.
En esta charla, presentaré una motivación de por qué estas álgebras interesan tanto a geómetras como analistas, y explicaré algunos de los resultados que hemos obtenido con Diego Lagos (Universidad de La Frontera) al respecto de una construcción reciente que asocia álgebras de Lie nilpotentes gradadas de paso dos a grafos dirigidos etiquetados. Podemos, a través del estudio del grafo dirigido, caracterizar ciertas sub-álgebras e ideales de estas, usando herramientas básicas de teoría espectral de grafos.
2022-10-11
16:00hrs.
16:00hrs.
Nikola Kamburov. PUC Santiago
Nondegeneracy and curvature bounds for stable free boundaries
Sala 2, edificio Rolando Chuaqui
Abstract:
One of the basic estimates on which the regularity theory of energy-minimizing free boundaries rests on is the so-called nondegeneracy bound. In the context of the one-phase free boundary problem (FBP) it says that a solution that is a minimizer of the underlying energy functional grows linearly away from the free boundary.
In this talk I will present our result that the nondegeneracy bound holds more broadly for stable solutions of the one-phase FBP, that is, for critical points of the energy functional, having non-negative second variation. As an application, we obtain local curvature estimates for stable free boundaries in dimension $n$, provided that the
Bernstein-type theorem for stable, entire solutions in the same dimension is valid. In particular, we get this curvature estimate in $n=2$ dimensions.
This is joint work with Kelei Wang (Wuhan University).
Nondegeneracy and curvature bounds for stable free boundaries
Sala 2, edificio Rolando Chuaqui
Abstract:
One of the basic estimates on which the regularity theory of energy-minimizing free boundaries rests on is the so-called nondegeneracy bound. In the context of the one-phase free boundary problem (FBP) it says that a solution that is a minimizer of the underlying energy functional grows linearly away from the free boundary.
In this talk I will present our result that the nondegeneracy bound holds more broadly for stable solutions of the one-phase FBP, that is, for critical points of the energy functional, having non-negative second variation. As an application, we obtain local curvature estimates for stable free boundaries in dimension $n$, provided that the
Bernstein-type theorem for stable, entire solutions in the same dimension is valid. In particular, we get this curvature estimate in $n=2$ dimensions.
This is joint work with Kelei Wang (Wuhan University).
2022-10-04
16:00hrs.
16:00hrs.
Carlos Román . PUC Santiago
Vórtices en el modelo de superconductividad de Ginzburg-Landau, lo viejo y lo nuevo
Sala 2, edificio Rolando Chuaqui
Abstract:
https://us06web.zoom.us/j/82129510934
Vórtices en el modelo de superconductividad de Ginzburg-Landau, lo viejo y lo nuevo
Sala 2, edificio Rolando Chuaqui
Abstract:
La superconductividad es un fenómeno que ha atraído muchísima atención desde su descubrimiento en 1911 por Onnes. Dos de sus características más llamativas son la posibilidad de circulación de corrientes eléctricas sin disipación y la levitación superconductora mediante la expulsión de un campo magnético aplicado.
En 1950 Ginzburg y Landau propusieron un modelo fenomenológico para su estudio, el cual ha sido tremendamente exitoso, con varios premios Nobel otorgados por su análisis. En presencia de un campo magnético aplicado, este modelo predice exitosamente la aparición en un superconductor de tipo II de defectos topológicos cuantizados denominados vórtices (similares a los de dinámica de fluidos). En esta charla comenzaremos por describir el comportamiento de superconductores de tipo II (y de sus correspondientes vórtices) en diferentes regímenes de intensidad de un campo magnético aplicado en 2D, y posteriormente presentaremos los últimos avances en el análisis del modelo en 3D.
https://us06web.zoom.us/j/82129510934
2022-09-20
16:00hrs.
16:00hrs.
Felix Schulze. University of Warwick
Neck pinches along the Lagrangian mean curvature flow of surfaces
Sala 2, edificio Rolando Chuaqui
Abstract:
Let L_t be a zero Maslov, rational Lagrangian mean curvature flow in a compact Calabi–Yau surface, and suppose that at the first singular time a tangent flow is given by the static union of two transverse planes. We show that in this case the tangent flow is unique, and that the flow can be continued past the singularity as an immersed, smooth, zero Maslov, rational Lagrangian mean curvature flow. This is joint work with Jason Lotay and Gábor Székelyhidi.
Neck pinches along the Lagrangian mean curvature flow of surfaces
Sala 2, edificio Rolando Chuaqui
Abstract:
Let L_t be a zero Maslov, rational Lagrangian mean curvature flow in a compact Calabi–Yau surface, and suppose that at the first singular time a tangent flow is given by the static union of two transverse planes. We show that in this case the tangent flow is unique, and that the flow can be continued past the singularity as an immersed, smooth, zero Maslov, rational Lagrangian mean curvature flow. This is joint work with Jason Lotay and Gábor Székelyhidi.
2022-08-23
16:00hrs.
16:00hrs.
Esteban Paduro. PUC
Ill-posedness Results for the Muskat problem and other Fluid equations
Sala 2, edificio Rolando Chuaqui
Abstract:
In the first part of this talk I will introduce some strategies to study ill-posedness and explain how these results contribute to the understanding of the well-posedness for different fluid equations. In the second part we will look in more detail to a particular ill-posedness result for a problem known as the Muskat equation. This result establishes that for a certain sequence of approximations of the Muskat equation obtained via Taylor expansion, their corresponding solution maps are not C^2 in some supercritical spaces that approach a critical one. This is done by considering some highly oscillatory initial data with small norm and showing that the map to the second Picard's iterate exhibits norm inflation which imply discontinuity of the solution map around the origin.
Ill-posedness Results for the Muskat problem and other Fluid equations
Sala 2, edificio Rolando Chuaqui
Abstract:
In the first part of this talk I will introduce some strategies to study ill-posedness and explain how these results contribute to the understanding of the well-posedness for different fluid equations. In the second part we will look in more detail to a particular ill-posedness result for a problem known as the Muskat equation. This result establishes that for a certain sequence of approximations of the Muskat equation obtained via Taylor expansion, their corresponding solution maps are not C^2 in some supercritical spaces that approach a critical one. This is done by considering some highly oscillatory initial data with small norm and showing that the map to the second Picard's iterate exhibits norm inflation which imply discontinuity of the solution map around the origin.
2022-08-09
14:00hrs.
14:00hrs.
Javier Castro. Universidad de Chile
Kolmogorov equation via deep learning methods
Sala 2, Departamento de Matematicas (edificio Rolando Chuaqui)
Kolmogorov equation via deep learning methods
Sala 2, Departamento de Matematicas (edificio Rolando Chuaqui)
2022-07-19
16:00hrs.
16:00hrs.
Jessica Trespalacios. Universidad de Chile, Dim
Existencia Global y Comportamiento a Largo Plazo del Modelo Quiral Principal 1+1 dimensional con Aplicaciones a Solitones
Sala 2, edificio Rolando Chuaqui
Abstract:
Consideramos el modelo de campo quiral principal (PCF) en 1+1 dimensiones de valor vectorial, obtenido como una simplificación de las ecuaciones de campo de Einstein en el vacío bajo la simetría Belinski-Zakharov. El modelo PCF es un modelo integrable, pero una descripción rigurosa de su evolución está lejos de ser completa. Aquí proporcionamos la existencia de soluciones locales en un espacio de energía adecuado, así como soluciones pequeñas globales suaves bajo una cierta condición de no degeneración. También construimos funcionales viriales que proporcionan una clara descripción del decaimiento de las soluciones globales suaves dentro del cono de luz. Finalmente, se presentan algunas aplicaciones en el caso de solitones del modelo PCF, un primer paso hacia el estudio de su estabilidad no lineal.
Existencia Global y Comportamiento a Largo Plazo del Modelo Quiral Principal 1+1 dimensional con Aplicaciones a Solitones
Sala 2, edificio Rolando Chuaqui
Abstract:
Consideramos el modelo de campo quiral principal (PCF) en 1+1 dimensiones de valor vectorial, obtenido como una simplificación de las ecuaciones de campo de Einstein en el vacío bajo la simetría Belinski-Zakharov. El modelo PCF es un modelo integrable, pero una descripción rigurosa de su evolución está lejos de ser completa. Aquí proporcionamos la existencia de soluciones locales en un espacio de energía adecuado, así como soluciones pequeñas globales suaves bajo una cierta condición de no degeneración. También construimos funcionales viriales que proporcionan una clara descripción del decaimiento de las soluciones globales suaves dentro del cono de luz. Finalmente, se presentan algunas aplicaciones en el caso de solitones del modelo PCF, un primer paso hacia el estudio de su estabilidad no lineal.
2022-07-12
13:00hrs.
13:00hrs.
David Padilla-Garza. Tu Dresden
A homogenized bending theory for prestrained plates
zoom https://zoom.us/j/9190316751
Abstract:
In this talk, we derive an effective bending plate model via simultaneous homogenization and dimension reduction. Our starting point is a 3d nonlinear elasticity model describing a composite whose components are prestrained with a magnitude that scales with the thickness of the plate. We assume that both the composite as well as the prestrain feature a periodic microstructure. After deriving the effective model via Gamma-convergence, we specialize in a class of examples that are explicitly solvable using a combination of analysis and numerics. Within this class of examples, we find several interesting and counterintuitive phenomena.
A homogenized bending theory for prestrained plates
zoom https://zoom.us/j/9190316751
Abstract:
In this talk, we derive an effective bending plate model via simultaneous homogenization and dimension reduction. Our starting point is a 3d nonlinear elasticity model describing a composite whose components are prestrained with a magnitude that scales with the thickness of the plate. We assume that both the composite as well as the prestrain feature a periodic microstructure. After deriving the effective model via Gamma-convergence, we specialize in a class of examples that are explicitly solvable using a combination of analysis and numerics. Within this class of examples, we find several interesting and counterintuitive phenomena.
2022-06-07
16:00hrs.
16:00hrs.
Humberto Prado . Universidad de Santiago de Chile
Fractional Pseudo-Differential Operators
Sala 2, edificio Rolando Chuaqui
Abstract:
The aim of this talk is to present some new results concerning the qualitative properties of a class of pseudo-differential equations. The motivation to study this class of equations originates in the mathematical physics literature.
Fractional Pseudo-Differential Operators
Sala 2, edificio Rolando Chuaqui
Abstract:
The aim of this talk is to present some new results concerning the qualitative properties of a class of pseudo-differential equations. The motivation to study this class of equations originates in the mathematical physics literature.
2022-05-31
16:00hrs.
16:00hrs.
Cesar Arias. PUC
Geometría y Holografía
Sala 2, edificio Rolando Chuaqui
Abstract:
Revisaremos las diferentes texturas geométricas que aparecen en la formulación más simple (en el límite de baja energía) de la correspondencia AdS/CFT. En particular, mostraremos como técnicas en geometría conforme pueden ser usadas para el cálculo de anomalías e invariantes conformes. Finalmente propondremos una extensión de las ideas anteriores, la cual surge naturalmente al tratar defectos como bordes generalizados.
Geometría y Holografía
Sala 2, edificio Rolando Chuaqui
Abstract:
Revisaremos las diferentes texturas geométricas que aparecen en la formulación más simple (en el límite de baja energía) de la correspondencia AdS/CFT. En particular, mostraremos como técnicas en geometría conforme pueden ser usadas para el cálculo de anomalías e invariantes conformes. Finalmente propondremos una extensión de las ideas anteriores, la cual surge naturalmente al tratar defectos como bordes generalizados.
2022-05-10
16:00hrs.
16:00hrs.
Gabrielle Nornberg . Departamento de Ingeniería Matemática, Universidad de Chile
Propiedades cualitativas para sistemas de tipo Lane-Emden
Sala 2, edificio Rolando Chuaqui
Abstract:
En esta charla discutiremos algunas propiedades cualitativas de soluciones de sistemas de tipo Lane-Emden y sus aplicaciones a la existencia y no existencia de soluciones.
Propiedades cualitativas para sistemas de tipo Lane-Emden
Sala 2, edificio Rolando Chuaqui
Abstract:
En esta charla discutiremos algunas propiedades cualitativas de soluciones de sistemas de tipo Lane-Emden y sus aplicaciones a la existencia y no existencia de soluciones.
Contáctenos
Edificio Rolando Chuaqui, Campus San Joaquín. Avda. Vicuña Mackenna 4860, Macul, Chile.Teléfono (+56) 95504 4511.
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