# 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: Pilar Herreros y Tai Nguyen

2018-04-24
16:00hrs.
Kirill Cherednichenko. University of Bath
Dispersive Effective Behaviour of High-Contrast Periodic Media
Sala 2 Facultad de Matemáticas
Abstract:
I will discuss my recent work with Y. Ershova and A. Kiselev, demonstrating that spectral problems for quantum graphs with rapidly oscillating high-contrast weights are asymptotically equivalent to "homogenised'' models with energy-dependent interface conditions. We show that these asymptotically equivalent models are directly related (in the sense of Schur-Frobenius duality) to models for time-dispersive media, which in the time domain involve memory, and we characterise the corresponding time convolution kernels explicitly.
2018-04-10
16:00hrs.
Daniel Alvarez-Gavela. Stanford University
The Simplification of Caustics
Sala 2 Facultad de Matemáticas
Abstract:
When light is reflected or refracted by a curved object, it accumulates on a caustic curve which typically has isolated semi-cubical cusp singularities. We will describe an h-principle technique that allows for the simplification of more complicated wavefront singularities into superpositions of the familiar semi-cubic cusp.
2017-11-28
16:00hrs.
Peter Veerman. Portland State University
Strange Convex Sets
Sala 2, Facultad de Matemáticas
Abstract:
Given a closed convex set $\Omega \in R^n$ , the metric projection of a given point $x ∈ R^n$ is given by the unique point $\Pi(x) ∈ \Omega$ that minimizes the (Euclidean) distance $\{|y − x| | y ∈ \Omega\}$ between $\Omega$ and $x$. Most mathematicians tend to think of convex sets in R n as very tame objects. It is therefore surprising that it is easy to construct a compact convex set $\omega$ in $R^2$ with the following strange property [Shapiro, 1994]: There is a point $x \notin \Omega$ and a vector $v$ such that the directional derivative
$$\lim_{t\to 0}\frac{ \Pi(x+vt)-?pi(x)}{t}$$
fails to exist. Note that for example convex polygons are not strange in this sense.

We revisit and modify that construction to obtain a convex curve in $R^2$ that is $C^{1,1}$ or differentiable with Lipschitz derivative, and that this curve bounds a convex set that has the property that the directional derivative of the projection is not defined. We also show how this construction can be made $C^n$ for n ≥ 2 except at a single point, and such that directional differentiability still fails.
2017-11-14
16:00hrs.
Eiji Yanagida. Tokyo Institute of Technology
Dynamics of Interfaces in The Fisher-Kpp Equation for Slowly Varying Initial Data
Sala 2, Facultad de Matemáticas
2017-10-31
16:00hrs.
Marie-Françoise Bidaut-Véron. Université de Tours, France
A Priori Estimates and Initial Trace for a Hamilton-Jacobi Equation With Gradient Absorption Terms
Sala 2, Facultad de Matemáticas
2017-10-24
16:00hrs.
Laurent Véron. Université François-Rabelais, Tours, France
Separable P-Harmonic Functions in a Cone, $1 < P \leq \infty$
Sala 2, Facultad de Matemáticas
2017-10-03
16:00hrs.
Ignacio Guerra . Usach
Multiplicty of Solutions for An Elliptic Equation With a Singular Nonlinearity and a Gradient Term
Sala 2, Facultad de Matemática
2017-09-05
16:00hrs.
Leonelo Iturriaga. Universidad Técnica Federico Santa María
Teoremas de Liouville Para Soluciones Radiales de Ecuaciones Elípticas Semilinales
Sala 2, Facultad de Matemática
Abstract:
En esta charla presentaremos algunos teoremas de Liouville para soluciones positivas radialmente simétricas de la ecuación
$$-\Delta u=f(u) \ \text{ en } \mathbb{R}^n$$
donde $f$ es una función continua en $[0,+\infty)$ que es positiva en $(0,+\infty)$. Nuestro enfoque nos permite considerar problemas mas generales, donde la no linealidad puede ser multiplicada por un peso radialmente simétrico y/o el Laplaciano es reemplazado por el $p$-Laplaciano, $1 < p < N$.
2017-06-06
16:00hrs.
Pilar Herreros. PUC
Mediatrices en Superficies
Sala 2, Facultad de Matemática
Abstract:
Dados dos puntos en una superficie Riemanniana llamamos mediatriz al conjunto de puntos equidistante a ambos puntos dados.

Hablaremos de la estructura que tiene este conjunto y de su regularidad,  en particular de que sus vertices son radialmente linearizables.

2017-05-30
16:00hrs.
Monica Musso. PUC
Existence, Compactness and Non Compactness for Fractional Yamabe Problem
Sala 2, Facultad de Matemática
Abstract:
Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n, [h])$. The fractional Yamabe problem consists in finding a metric in the conformal class $[h]$ whose fractional scalar curvature is constant.

In this talk, I will present some recent results concerning existence of solutions to the fractional Yamabe problem,  and also properties of compactness and non compactness of its solution set, in comparison with what is known in the classical case.

These results are in collaboration with Seunghyeok Kim and Juncheng Wei.
2017-05-23
16:00hrs.
Duvan Henao. PUC
Biaxiality in Liquid Crystals At Low Temperatures
Sala 2, Facultad de Matemática
Abstract:
I will present a joint work with Apala Majumdar and Adriano Pisante where we study the Nobel-winner Landau-de Gennes functional for nematic liquid crystals. We identify the location of defects in the low temperature limit and show the coexistence of biaxial and negative uniaxial points around each defect. We also estimate the size of the biaxial regions.
2017-05-09
16:00hrs.
Suspendido. PUC
Suspendido
Sala 2, Facultad de Matemáticas UC
2017-05-02
16:00hrs.
Hanne Van Den Bosch. PUC
Spectrum of Dirac Operators Describing Graphene Quantum Dots
Sala 2, Facultad de Matemáticas UC
Abstract:
Low energy electronic excitations in graphene, a two-dimensional lattice of carbon atoms, are described effectively by a two–dimensional Dirac operator. For a bounded flake of graphene (a quantum dot), the choice of boundary conditions determines various properties of the spectrum. Several of these choices appear in the physics literature on graphene. For a simply connected flake and a family of boundary conditions, we obtain an explicit lower bound on the spectral gap around zero. We can also study the effect of the boundary conditions on eigenvalue sums in the semiclassical limit. This is joint work with Rafael Benguria, Søren Fournais and Edgardo Stockmeyer.
2017-04-25
16:00 hrs.
Carmen Cortázar. PUC
Large Time Behavior of Porous Medium Solutions in Exterior Domains
Sala 2, Facultad de Matemáticas UC
Abstract:
Let  $\mathcal{H}\subset \mathbb{R}^N$ be a non-empty bounded open set. We consider the porous medium equation in the complement of  $\mathcal{H}$ ,  with zero Dirichlet data on  its boundary and nonnegative compactly supported integrable initial data.

Kamin and Vázquez, in 1991, studied the large time behavior of solutions of such  problem  in space dimension 1.    Gilding and Goncerzewicz, in 2007, studied this same problem  dimension 2.  However, their result does not say much about the behavior when the points  are in the so called near field scale. In particular, it does not give a sharp decay rate, neither a nontrivial asymptotic profile, on compact sets.
In this paper we characterize the large time behavior in such scale, thus completing their results.

This a Joint work with Fernando Quiros ( Universidad Autonoma de Madrid, Spain) and Noemí Wolanski ( Universidad de Buenos Aires, Argentina).

2017-04-18
16:00hrs.
Abraham Solar. PUC
Stability of Semi-Wavefronts for Delayed Reaction-Diffusion Equations
Sala 2, Facultad de Matemáticas UC
Abstract:
Semi-wavefronts are bounded positive solutions of delayed reaction-diffusion equations such that its shape is not changed in the time and they move with constant speed en the time. In this talk I will show the most important results about stability of this solutions and how they determinate the propagation speed of the a broad class of solutions.
2017-04-04
16:00hrs.
Erwan Hingant. Universidad del Bio-Bio
The Stochastic Becker-Döring System
Sala 2, Facultad de Matemáticas UC
Abstract:
The Becker-Döring equations might be "one of the simplest kinetic model to describe a number of issues in the dynamics of fase transitions", Penrose (1989). This model describes the evolution of the concentration of clusters (or aggregates) according to their size. The rules are simple, a cluster of size $i$ may encounter a particle (cluster of size $1$) to form a new one of size $i+1$. Conversely, a cluster of size $i$ could release a particle leading to a cluster of size $i-1$. In this talk we will present the stochastic version of this rules when the system consists in a finite number of particles, namely a pure jump Markov process on a finite state space. And we will discuss about some results and issues around the law of large number associate to this problem.

Ref.: E. Hingant and R. Yvinec, Deterministic and Stochastic Becker-Döring equations: Past and Recent Mathematical Developments, Preprint arXiv:1609.00697, 2016.
2017-03-28
16:00hrs.
Sophia Jahns . Tübingen University, Germany
Trapped Light in Stationary Spacetimes
Sala 2, Facultad de Matemáticas UC
Abstract:
Light can circle a massive object (like a black hole or a neutron star) at a "fixed distance", or, more generally, circle the object without falling in or escaping to infinity. This phenomenon is called trapping of light and well understood in static, asymptotically flat (AF) spacetimes. If we drop the requirement of staticity, similar behavior of light is known, but there is no definiton of trapping available. We present some known results about trapping  of light in static AF spacetimes. Using the Kerr spacetime as a model, we then show how trapping can be better understood in the framework of phase space and work towards a definition for photon regions in stationary AF spacetimes.
2017-03-21
16:00hrs.
Mauricio Bogoya. Universidad Nacional de Colombia, Bogota
A Non-Local Diffusion Coupled System Equations in a Bounded Domain
Sala 2, Facultad de Matemáticas UC
2017-03-14
16:00hrs.
Marcos de la Oliva. Universidad Autónoma de Madrid
Relaxation of a Model for Nematic Elastomers
Sala 2, Facultad de Matemáticas UC
Abstract:
The direct method of the calculus of variations to find minimizers is based on compactness and lower semicontinuity of the energy functional. In the absence of lower semicontinuity, one option is to find the relaxation, i.e., the largest lower semicontinuous functional below a given one. In nonlinear elasticity, computing the relaxation is difficult beacuse of the non-standard growth conditions. In this talk we show that the relaxation for a model in nonlinear elasticity is given by the quasiconvexification of the integrand. We also propose a model for nematic elastomers (a kind of liquid crystals) in which the energy has a part in the reference configuration and a part in the deformed configuration. We show again that the relaxation is given by the quasiconvexification.
2017-01-11
15:00hrs.
Marie-Françoise Bidaut-Véron. Université François Rabelais, Tours, France
A Priori Estimates and Ground States of Solutions of An Emden-Fowler Equation With Gradient
Sala 2, Facultad de Matemáticas UC