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

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.
 
2018-11-20
16:00hrs.
Natham Aquirre Quiñonez. PUC
Funciones p-armónicas con condiciones de Neumann en el borde que involucran medidas
Sala 2
Abstract:
En esta presentación discutiré el problema de encontrar funciones p-armónicas en el semi-espacio superior con condiciones de Neumann en el borde de tipo no lineal y con medidas. Para ello introduciré el concepto de soluciones renormalizadas en dominios acotados y explicaré como usar esta teoría para obtener soluciones de nuestro problema. La idea principal es usar estabilidad local y la simetría del dominio (y el operador). Luego aplicaremos estas ideas para obtener resultados de existencia para diferentes tipos de no linealidades. Las técnicas a utilizar también nos permitirán establecer algunos resultados de no existencia y de eliminación de singularidades.
2018-11-19
16:00hrs.
Natham Aquirre Quiñonez. P. Universidad Católica de Chile
Funciones p-armónicas con condiciones de Neumann en el borde que involucran medidas
Sala 2
Abstract:
En esta presentación discutiré el problema de encontrar funciones p-armónicas en el semi-espacio superior con condiciones de Neumann en el borde de tipo no lineal y con medidas. Para ello introduciré el concepto de soluciones renormalizadas en dominios acotados y explicaré como usar esta teoría para obtener soluciones de nuestro problema. La idea principal es usar estabilidad local y la simetría del dominio (y el operador). Luego aplicaremos estas ideas para obtener resultados de existencia para diferentes tipos de no linealidades. Las técnicas a utilizar también nos permitirán establecer algunos resultados de no existencia y de eliminación de singularidades.
2018-11-13
16:00hrs.
Jose Gabriel Torres. Pontificia Universidad Católica de Chile
Solitones por traslación de una familia de flujos no degenerados
Sala 2
Abstract:
Los flujos por curvatura extrínseca estudian la evolución de superficies en ambientes (semi-)riemannianos, donde la velocidad del flujo está dada por una función en las curvaturas principales sobre la evolución de la superficie. Dicha evolución esta determinada por una ecuación de tipo parabólico no lineal. En esta charla me concentraré en una familia de flujos donde la función velocidad está dada por un cociente entre polinomios simétricos elementales con especial énfasis en soluciones auto-similares donde esta que actúa por traslación. Discutiré resultados conocidos y futuras lineas de trabajo en este contexto.
2018-10-30
16:00hrs hrs.
Michal Kowalczyk. Departamento de Ingeniería Matemática Universidad de Chile
Maximal solution of the Liouville equation in doubly connected domains
Sala 2
Abstract:

 
 
   
In this talk I will discuss a new existence result for the widely  studied Liouville problem $\Delta u+\lambda^2 e^{\,u}=0$ in a bounded, two dimensional, doubly connected  domain with Dirichlet boundary conditions. I will show that for a sequence of   $\lambda_n\to 0$ this equation has solutions that blow-up in  in the whole domain.  Profiles of the blowing-up solutions are related to a free boundary problem which gives a solution to an optimal partition problem for the given domain. I will also describe the role of  the free boundary problem in  other classical equations such as the mean field  model or the prescribed Gaussian curvature equation.
 
2018-10-23
16:00hrs.
Mircea Petrache. Pontificia Universidad Católica de Chile
Desigualdad isoperimétrica en grafos regulares y formas de unos cristales
Sala 2
Abstract:
Voy a presentar unas técnica clásica para demostrar desigualdades isoperimétricas en R^n respecto de varias nociones de perímetro.
 Después vamos a ver cómo las mismas técnicas se pueden tal vez transferir a problemas combinatorios en unos grafos periódicos, y permiten predicir la forma de unos cristales.
 
2018-10-09
15:30hrs.
Mariel Sáez. Pontificia Universidad Católica de Chile
Sobre la unicidad del flujo de curvatura media en graficos de funciones
Sala 2
Abstract:

Voy a discutir resultados recientes con P. Daskalopoulos de condiciones suficientes para la unicidad de soluciones de la ecuación de evolución asociada a gráficos de funciones evolucionando por flujo de curvatura media. Compararemos estos resultados con el comportamiento de soluciones clásicas a la ecuación del calor.
 
2018-09-25
16:00hrs.
Nikola Kamburov. Pontificia Universidad Católica de Chile
On positive solutions of the Lane-Emden equation in the plane
Sala 2 FMAT
Abstract:
We prove that positive solutions of the Lane-Emden equation in a two-dimensional smooth bounded domain are uniformly bounded for all large exponents. Recent work of De Marchis, Grossi, Ianni and Pacella 
provides a fairly complete asymptotic description of such solutions, under a certain integral bound condition. Furthermore, they establish the asymptotic uniqueness of positive solutions satisfying that bound in convex planar domains. We remove this condition by showing that the bound is always satisfied in star-shaped domains.

This is joint work with Boyan Sirakov (PUC-Rio).
2018-09-04
16:00hrs.
Matías Courdurier. Pontificia Universidad Católica de Chile
Construction of Solutions for some Localized Nonlinear Schrodinger Equations
Sala 2
Abstract:
In this talk we will present the constructions of solution of the following reduced non-linear Schrodinger equation: -u''+V(x)h'(|u|^2)u = w u, where V(x)=1 for |x|<1 and 0 otherwise, and where h' is any continuous
strictly increasing function. Reduced non-linear Schrodinger equation are important as mean-field approximations of quantum systems and the constructed solutions characterize bound-states of the dynamic version of the equation.