Seminario de Análisis y Geometría

Los seminarios de Análisis y Geometría se llevan a cabo los días martes a las 16:00 en la sala 2 de la Facultad de Matemáticas, Pontificia Universidad Católica de Chile.

Organizadores: Marta García Huidobro

2020-08-25
17:10hrs.
Nicolás Vilches. Pontificia Universidad Católica de Chile
Invertibilidad global en espacios de Sobolev: parte II
Zoom, ID de reunión: 914 0230 0356 Código de acceso: Laplace
Abstract:

En esta charla nos enfocaremos en una de las herramientas mencionadas durante la parte I, respecto a cómo recuperar el determinante de la matriz jacobiana de manera distribucional. Estudiaremos una conjetura propuesta por John Ball en 1976, junto con un ejemplo que ilustra la posibilidad de tener un determinante distribucional distinto al puntual. Posteriormente, seguiremos la demostración de Stefan Müller a la conjetura (en 1990), a partir de un resultado más general. La herramienta principal será una versión refinada del teorema de diferenciación de Lebesgue, debida a Alberto Calderón y Antoni Zygmund.

 
2020-08-18
17:10hrs.
Duvan Henao. Pontificia Universidad Católica de Chile
Invertibilidad global en espacios de Sobolev: parte I
Zoom Meeting ID: 980 2662 5924 Passcode: Brouwer  
Abstract:
La charla trata sobre la regularidad requerida para definir el grado topológico en espacios de Sobolev, a modo de mantener una de sus propiedades esenciales: si el grado de una función es igual a uno, con respecto a un punto y del espacio, restringiendo la función a una subregión E del dominio, entonces el punto y tiene exactamente una preimagen en la subregión E. Esto se usa en elasticidad para garantizar que no haya interpenetración de la materia (al menos para el problema Dirichlet donde el desplazamiento se prescribe en toda la frontera). Junto a Carlos Mora Corral y a Marcos de la Oliva hemos relajado las condiciones de regularidad para que se mantenga esa propiedad del grado. El análisis está conectado con la estructura analítica de los menores de un gradiente (en particular la identidad de Piola) y la rigidez geométrica. Las ideas presentadas podrían ayudar a establecer que ciertas funciones son difeomorfismos, incluso en contextos con variedades diferenciales de dimensión muy grande.
https://zoom.us/j/98026625924?pwd=WE5yaXRjNDc1S2xMQXFsdThBaXhrQT09
2020-03-13
15:00hrshrs.
Karina Vilches. Universidad Católica del Maule
Emergent behaviors in multi-cellular tumor progression including micro-environmental interactions anunciado. Atención: Seminario suspendido por razones de fuerza mayor
Sala 1, Facultad de Matemáticas
Abstract:
Atención: Seminario suspendido por razones de fuerza mayor

We present a mathematical approach that captures and explores a wide range of mechanisms and biological variability in tumor progression to better understand the orchestrate multiple phenomena in cancer dynamics. In this respect, Mathematical Biology is needed to promote the realization of modeling platforms that facilitate the discovery of novel biological phenomena, rules, and theories. Therefore, the main goal of this presentation corresponds to discuss the analysis of a mathematical model that represents a multi-cellular chemotaxis-haptotaxis interaction in Cancer progression. The main novelty consists in applying the non-linear analysis of parabolic-elliptic system and numerical approximation to describe the micro-environment effects over tumor progression.
2020-01-21
16:00hrshrs.
Barbara Brandolini. Departamento de Matemáticas, Universidad de Nápoles, Italia
Improved bounds for Hermite-Hadamard inequalities in higher dimensions
Sala 2, Facultad de Matemáticas
Abstract:
ver pdf
2019-12-03
16:00hrs.
Rafael Benguria. P. Universidad Católica de Chile
A General Brezis-Nirenberg Problem
Facultad de Matemáticas, sala 2
Abstract:
In this talk I will discuss existence, nonexistence and uniqueness of solutions for a general Brezis-Nirenberg problem. The region of parameters for which there is existence and nonexistence of positive solutions is characterized by two spectral problems that have interest on their own. This is joint
work with Soledad Benguria (U. Wisconsin, Madison).
2019-11-05
16:30hrs.
Francois Murat. Laboratoire Jacques-Louis Lions Sorbonne Université & Cnrs
Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a singularity at u = 0
Facultad de Matematicas, sala 2
2019-10-08
16:00hrs.
Marcone Pereira. Universidad de Sao Paulo
Nonlocal equations in perforated domains
Sala 2
Abstract:
In this talk, we analyze the asymptotic behavior of nonlocal problems widely used in the modeling of diffusion or dispersion processes. We consider an integral-differential equation, with nonsingular kernel, in a limited domain Ω from which we remove subsets that we call holes. We deal with Neumann and Dirichlet conditions in the holes setting Dirichlet outside of Ω. Assuming the weak convergence of the family of functions which represents such holes, we analyze the limit of the solutions of the equations obtaining the existence of a limit problem. In the case where the holes are removed periodically, we observe that the critical radius is of order of the typical cell size (which gives the period). Finally we study the behavior of these problems when we resize their kernel with the objective of approaching local partial differential equations discussing peculiarities.
2019-09-24
16:00hrs.
Renato Velozo. Universidad de Cambridge, Uk.
Gravitational collapse for the Einstein-Vlasov system with spherical symmetry
Sala 2
Abstract:
The Einstein's equations form a model of the spacetime which has been widely studied in general relativity. This model studies the evolution on time of the matter and a Lorentzian metric which together shape the spacetime. In this talk, we will study specifically the Einstein-Vlasov system, a particular case of the Einstein's equations where the matter is modeled through a distribution of matter as it is usual in kinetic theory. Firstly, I will present some of the main features about the full Einstein-Vlasov system. Secondly, I will present some results about gravitational collapse for the Einstein-Vlasov system with spherical symmetry proved by M. Dafermos and A. Rendall in 2006. Finally, I will mention some problems which are part of my current research.
2019-08-27
16:00hrshrs.
Carlos Román. Pontificia Universidad Católica de Chile
On the 3D Ginzburg-Landau model of superconductivity.
Sala 2, Facultad de Matemáticas
Abstract:
The Ginzburg-Landau model is a phenomenological description of superconductivity. A crucial feature is the occurrence of vortices (similar to those in fluid mechanics, but quantized), which appear above a certain value of the strength of the applied magnetic field called the first critical field. In this talk I will present a sharp estimate of this value and describe the behavior of global minimizers for the 3D Ginzburg-Landau functional below and near it. This is partially joint work with Etienne Sandier and Sylvia Serfaty.
2019-08-20
16:00hrs.
Bruno Premoselli. Universidad Libre de Bruselas
Examples of Compact Einstein four-manifolds with negative curvature
Sala 2, Facultad de Matemáticas UC
Abstract:
We construct new examples of closed, negatively curved
Einstein four-manifolds. More precisely, we construct  Einstein metrics
of negative sectional curvature on ramified covers of compact hyperbolic
four-manifolds with symmetries, initially considered by Gromov and
Thurston. These metrics are obtained through a deformation procedure.
Our candidate approximate Einstein metric is an interpolation between a
black-hole Riemannian Einstein metric near the branch locus and the
pulled-back hyperbolic metric. We then deform it into a genuine solution
of Einstein’s equations, and the deformation relies on an involved
bootstrap procedure. Our construction yields the first example of
compact Einstein manifolds with negative sectional curvature which are
not locally homogeneous. This is a joint  work with J. Fine (ULB,
Brussels).
2019-06-18
16:00hrs.
Mircea Alexandru Petrache. Pontificia Universidad Católica de Chile
Distorted diffeomorphisms and the manifolds of speech and sound (part 2)
Sala 2, Facultad de Matemáticas
Abstract:
We continue with the proof of the results from arxiv:1610.08138v3 of Damelin, Fefferman and Glover, concerning an application of BMO theory and harmonic analysis towards dimensionality reduction algorithms for approximating high dimension speech and sound data by adapted low-dimensional manifolds.
2019-06-11
16:00hrshrs.
Mircea Alexandru Petrache. Pontificia Universidad Católica de Chile
Distorted diffeomorphisms and the manifolds of speech and sound, following Damelin-Fefferman-Glover
Sala 2, Facultad de Matemáticas
Abstract:
The aim of this introductory-level seminar is to present the results of the paper arxiv:1610.08138v3 of Damelin, Fefferman and Glover, concerning an application of BMO theory and harmonic analysis towards dimensionality reduction algorithms for approximating high dimension speech and sound data by adapted low-dimensional manifolds.
2019-05-07
16:00hrs.
Luciano Sciaraffia. PUC
Soluciones No Triviales a Problemas Sobredeterminados en Dominios Anulares
Sala 2, Facultad de Matemáticas
2019-04-16
16:00hrs.
Duván Henao. Pontificia Universidad Católica de Chile
Aplicando la geometría diferencial para comprender la estructura de los cristales líquidos
sala 2, Facultad de Matemáticas, PUC
Abstract:
Veremos como el teorema de Liouville y los teoremas de Schoen-Uhlenbeck nos permiten demostrar que a bajas temperaturas los defectos de los minimizadores del funcional de Landau-de Gennes son necesariamente biaxiales. (En particular, a bajas temperaturas es falso que sus frustraciones topológicas las resuelvan derritiéndose, como se asume comúnmente.)
2019-04-02
16:00hrshrs.
Ariane Trescases. Cnrs Imt Toulouse
Quaternions in collective motion
Sala 2, Facultad de Matemáticas
Abstract:
We present a model for multi-agent dynamics based on rigid alignment. Each agent is described by its position and body attitude: it travels at a constant speed while trying to coordinate its solid orientation with the solid orientation of the neighboring agents. The body orientations are represented by unitary quaternions. We first introduce an individual based model in the spirit of the Vicsek model, enhanced with the body orientation dynamics. We then derive the corresponding kinetic model. From there we compute the hydrodynamical limit, leading to a self-organized hydrodynamical system based on quaternions.
2019-03-26
16:00hrs.
Satoshi Tanaka. okayama University of Science, Japan
Uniqueness of positive radial solutions of superlinear elliptic equations in annuli
Sala 2
Abstract:
This is a joint work with Naoki Shioji (Yokohama National University) and
Kohtaro Watanabe (National Defense Academy).

The Dirichlet problem
\begin{equation*}
 \left\{
  \begin{array}{cl}
   \Delta u + f(u) =0 &  \mbox{in} \ x \in A, \\[1ex]
    u=0 & \mbox{on} \ \partial A
  \end{array}
 \right.
\end{equation*}
is considered, where $A:=\{x\in {\bf R}^N : a< |x| <b$,\  $N \in {\bf N}$, $N \ge 2$, $0<a<b<\infty$,
$f \in C^1[0,\infty)$, $f(u)>0$ and $uf'(u) \ge f(u)$ for $u>0$.
Positive radial solutions are studied.
Hence the boundary value problem
$$u'' + \frac{N-1}{r} u' + f(u) = 0, \quad r \in (a,b); \qquad   u(a) = u(b) = 0$$
is considered.
Uniqueness results of positive solutions are shown.
2019-03-19
16:00hrs.
Azahara de la Torre Pedraza. University of Freiburg
On higher dimensional singularities for the fractional Yamabe problem
Sala 2
Abstract:
We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, for which we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional order ODE,
and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to  examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of  Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.
2019-03-12
16:00hrs.
Andrés Larraín-Hubach. University of Dayton, ohio
Conexiones auto-duales sobre espacios Taub-NUT
Sala 2
Abstract:
Las ecuaciones de Yang-Mills son un sistema de ecuaciones en derivadas parciales, definidas sobre variedades suaves en cuatro dimensiones, con un profundo significado geométrico. Las propiedades de las soluciones de estas ecuaciones, sobre variedades compactas, han sido  analizadas desde los años sesenta y han arrojado resultados importantes tanto en matemáticas como en física. Las soluciones sobre  variedades no compactas no han sido estudiadas tan ampliamente y aún hay muchas preguntas importantes sin respuesta. En esta charla, basada en resultados obtenidos en colaboración con Sergey Cherkis y Mark Stern, explicaré diversas propiedades de ciertas  soluciones a las ecuaciones de Yang-Mills, definidas sobre unas variedades abiertas especiales llamadas Espacios Taub-NUT. En particular, explicaré dos argumentos distintos para probar un teorema de índice necesario en la construcción.
 
2019-01-03
16:00hrs.
Armin Schikorra . University of Pittsburgh
Self-repulsive curvature energies for curves and surfaces: regularity theory and relation to harmonic maps
Sala 2
Abstract:
I will talk about a class of curvature energies for curves, the O'Hara energies, that are nonlocal in nature. In particular, I will present an approach for regularity theory of minimizers and critical points for these curves which is based on a relation to (fractional) harmonic maps. Then I will present some results towards attempts of generalizing this idea to surfaces.
2018-12-19
14:00hrs.
Frank Morgan . Williams College
Isoperimetric Problems with Density
Sala 2
Abstract:
A round soap bubble is the least-perimeter way to enclose a given volume of air. Similarly, the familiar double soap bubble that forms when two soap bubbles come together is the least-perimeter way to enclose and separate two given volumes of air. What if you give space a radial density that weights both perimeter and volume? The presentation will include open questions and recent results, some by undergraduates. Students welcome.