Speaker: Rodolfo Viera 

Pontificia Universidad Católica de Chile

Date: Tuesday, September 6, 2022 at 12 Santiago time

Abstract: En 1994, Gromov preguntó si toda red separada del plano \( X\subset\mathbb{R}^2 \) (i.e, un conjunto discreto y denso de una manera uniforme) es bi-Lipschitz equivalente al lattice est\’andar \(\mathbb{Z}^2 \) (i.e si X es bi-Lipschitz rectificable). Esto fue respondido de manera negativa por Burago y Kleiner, e independientemente por McMullen. Su demostración se basa en la existencia de una función de densidad \(\rho:[0,1]^2\to\mathbb{R}\) tal que \( 0<\inf\rho<\sup\rho<\infty \) y para la cual la ecuación

$$
Jac(f)=\rho\qquad a.e
$$

no tiene solución bi-Lipschitz \( f:[0,1]^2\to\mathbb{R}^2\). En esta charla veremos algunos resultados en esta línea, por ejemplo condiciones suficientes para asegurar la rectificabilidad de una red separada como consecuencia de la existencia de soluciones bi-Lipschitz para ciertas ecuaciones que involucran un jacobiano. También intentaremos pasar por otros resultados de no-rectificabilidad bajo condiciones más débiles que bi-Lipchitz.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Hanne Van Den Bosch

Speaker: Erwin Topp

Universidad de Santiago, Chile

Date: Thursday, August 25, 2022 at 16 p.m. Santiago time

Abstract: In this talk I will report some multiplicity results for large solutions of fractional Hamilton-Jacobi equations posed on a bounded domain, subject to exterior Dirichlet conditions. We construct large solutions using the method of sub and supersolutions, following the classical approach of J.M. Lasry and P.L. Lions for second-order equations with subquadratic gradient growth. We identify two classes of solutions: the one coming from the natural scaling of the problem; and a one-parameter family of solutions, different from the previous, which can be formally described as a lower-order perturbation of blow-up fractional harmonic functions. Joint work with Alexander Quaas and Gonzalo Dávila (UTFSM-Chile).

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Claudio Muñoz

Speaker: Nicolas Torres Escorza

Université Claude Bernard Lyon 1, France

Date: Tuesday August 23, 2022 at 12 Santiago time

Abstract: We introduce and study an extension of the classical elapsed time equation in the context of neuron populations that are described by the elapsed time since last discharge. In this extension we incorporate the elapsed since the penultimate discharge and we obtain a more complex system of integro-differential equations. For this new system we prove convergence to stationary state by means of Doeblin’s theory in the case of weak non-linearities in an appropriate functional setting, inspired by the case of the classical elapsed time equation. Moreover, we present some numerical simulations to observe how different firing rates can give different types of behaviors and to contrast them with theoretical results of both classical and extended models.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, Beauchef 851,  piso 7
Chair: Salomé Martínez

Speaker: Argenis Mendez 

Pontificia Universidad Católica de Valparaíso

Date: August 16, 2022 at 12 Santiago time

Abstract: In this talk, we will present some new results related to the regularity properties of the initial value problem (IVP) for the equation
\begin{equation}\label{eq1}
\left\{
\begin{array}{ll}
\partial_{t}u-\partial_{x_{1}}(-\Delta)^{\alpha/2} u+u\partial_{x_{1}}u=0, \quad 0< \alpha< 2, & \\
u(x,0)=u_{0}(x),x=(x_{1},x_{2},\dots,x_{n})\in \mathbb{R}^{n},n\geq 2,& t\in\mathbb{R}, \\
\end{array}
\right.
\end{equation}
where $(-\Delta)^{\alpha/2}$ denotes the $n-$dimensional fractional Laplacian.

In the case that \(\alpha=2,\) the equation is known as the Zakharov-Kuznetsov-(ZK) equation, Zakharov and Kuznetsov proposed it as a model to describe the propagation of ion-sound waves in magnetic fields in dimension n=3.

A property that enjoys the solutions of the ZK equation is Kato’s smoothing effect. Roughly speaking, the solution to the initial value problem is, locally, one derivative smoother (in all directions) in comparison to the initial data.

The goal of this talk is to show that despite the non-local character of the operator \((-\Delta)^{\frac{\alpha}{2}}\), the solution of the equation (IVP) is locally smoother. It becomes \(\frac{\alpha}{2}-\) smoother in all directions.

As a byproduct, we show the applicability of this result in establishing the propagation of localized regularity of the solutions of (IVP) in a suitable Sobolev space.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, Beauchef 851,  piso 7
Chair: Claudio Muñoz

Speaker: Natham Aguirre

Universidad de Chile

Date: August 9, 2022 at 12 Santiago time

Abstract: In this talk I will discuss the problem of finding solutions to some nonlinear elliptic equations with measure data. To this end I will introduce the concept of Renormalized Solutions, which is a very useful tool to solve both Dirichlet and Neumann problems. I will present some results in this area and also discuss some open problems.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Hanne Van Den Bosch

Speaker: Andrés Zúñiga

Universidad de O’Higgins

Date: July 6, 2022 at 12 Santiago time

Abstract: We will discuss a variant of a classical geometric minimization problem, known as the “nonlocal isoperimetric problem”, which arises from studies in Nuclear Physics by Gamow in the 1930’s. By introducing a density in the perimeter functional, we obtain features that differ substantially from existing results in the framework of the classical problem without densities. In the regime of “small” non-local contribution, we completely characterize the minimizer, in the case the density is a monomial radial weight. 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.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Claudio Muñoz

Speaker: Sergio Rica

Instituto de Física, Pontificia Universidad Católica de Chile

Date: June 28, 2022 at 12 Santiago time

Abstract: Despite 250 years of history, the nature of solutions of the Euler equations remains an open problem. To date, it is not known if general smooth initial conditions of the Euler equations with finite energy do or do not blow-up in finite time. I will review the approach initiated by Leray of self-similar blow-up solutions. Lastly, I will show that under some conditions an axisymmetric incompressible and inviscid flow presents a finite-time singularity. This singularity appears to be generic and robust for a wide number of finite energy initial conditions.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Michal Kowalczyk

Speaker: Gastón Vergara

Maynooth University, Ireland

Date: June 22, 2022 at 12 Santiago time

Abstract: En esta charla comenzaremos abordando algunas formulaciones recientes de las ecuaciones water-waves, para luego tomar ventaja de ellas y establecer algunos problemas de transmisión explícitos que describen interacciones fluido-estructuras. En una segunda parte estudiaremos el cómo bajo ciertas restricciones es posible obtener algunas generalizaciones de la ecuación de Cummins. Por último, mostraremos métodos con los cuales la comunidad que estudia la extracción de energía a partir de las olas del mar utiliza dichas ecuaciones integro-diferenciales para obtener energías limpias e infinitamente renovables.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Gabrielle Nornberg

Speaker: Hamed Barzegar 

University of Vienna, Austria

Date: June 14, 2022 at 12 Santiago time

Abstract: In this talk, I will give a short introduction to “mathematical cosmology” with a focus on the application of the kinetic theory in cosmology. As such, I will talk about Bianchi cosmologies, i.e., spatially homogeneous spacetimes that are governed by the Einstein equations which are coupled to massless collisionless (Vlasov) matter. Then, I will discuss their future attractors and show future stability of such models within Bianchi types I, II, and V symmetry class. The proof turns out to be more challenging compared to the corresponding massive case where the cosmological constant is absent, since the massless particles indicate less decay rates in the course of the expansion of the universe. The proof is based on an energy method for small initial data.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Paola Rioseco

Speaker: Paz Albares

Universidad de Salamanca, Spain

Date: June 7, 2022 at 12 Santiago time

Abstract: The Painlevé Property has proved to be a fruitful tool when it comes to identifying the integrability of nonlinear PDEs. The combination of this technique with the so-called singular manifold method offers an ideal framework to approach nonlinear integrable systems: it provides a systematic methodology to obtain the associated spectral problem, as well as a recursive procedure to determine soliton-like solutions. In this talk, we review the main characteristics of this setting, with applications on several examples related to Nonlinear Schrödinger equations, in which solutions as solitons and lumps are thoroughly discussed.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Claudio Muñoz