Category Archives: Past seminar

Undestanding the APS boundary condition for the zero modes of the Dirac operator

Speaker: Marie Fialová

Institute of Science and Technology Austria

Date: Monday, December 11th, 2023 at 12 Santiago time

Abstract: How many zero modes (states with zero energy) are there of the Dirac operator with magnetic field in two dimensions? This question was answered by Aharonov and Casher in 1979 for the case of plane. They showed that this number is given by the flux of the magnetic field, more precisely the integer part of it. Moreover, the zero modes are chiral, aligning with the direction of the magnetic field. We investigate the same problem for the case of a plane wih holes considering the Atiyah–Patodi–Singer (APS) boundary condition (BC). This BC was introduced by APS in their famous series of three papers on the index theorem on manifolds with boundary in 1970’s.

If the manifold has a product structure near the boundary this BC allows extending the zero modes as square integrable functions to a semi-infinite cylinder glued to the boundary. I will discuss some qualitative properties of the zero modes in case this product structure near the boundary is missing.

This talk is based on my PhD work supervised by Jan Philip Solovej and a current work in progress with Hanne Van Den Bosch.

Venue:  Sala de Seminarios, 5th floor, Beauchef 851 / Online via Zoom
Chair: Ricardo Freire

 

Singular limits arising in mechanical models of tumor growth

Speaker: Noemi David

Université Claude Bernard – Lyon 1 (France)

Date: Monday, December 4th, 2023 at 12 Santiago time

Abstract: The mathematical modelling of cancer has been increasingly applying fluid-dynamics concepts to describe the mechanical properties of tissue growth. The bio-mechanical pressure plays a central role in these models, both as the driving force of cell movement and as an inhibitor of cell proliferation. In this talk, I will present how it is possible to build a bridge between models that have different pressure-velocity or pressure-density relations. In particular, I will focus on the inviscid limit from a visco-elastic model to a Darcy’s law-based model, and the incompressible limit that links the latter to a Hele-Shaw free boundary problem with density constraint.

Venue:  Sala de Seminarios, 5th floor, Beauchef 851 / Online via Zoom
Chair: Ricardo Freire

 

Stability and instability for the fractional KdV equation in all spatial dimensions

Speaker: Oscar Riaño

Universidad Nacional de Colombia, Bogotá

Date: Monday, November 27th, 2023 at 12 Santiago time

Abstract: We will study the fractional Korteweg-de Vries equation (fKdV) in several spatial dimensions. This is a nonlinear dispersive model where dispersive effects are studied through the fractional Laplacian. The equation appears in contexts such as water waves, waves in stratified fluids, boundary-layer type shear flow, and ion-acoustic waves, among others.

We will focus on traveling wave solutions (or solitary waves) of fKdV and analyze questions of orbital stability/instability of such solutions. We will see that our results are valid for arbitrary dimen- sions, thus we recover and establish new results for the most studied case the one-dimensional fKdV.

Joint work with Svetlana Roudenko (Florida International University (FIU)).

Venue:  Sala de Seminarios, 5th floor, Beauchef 851 / Online via Zoom
Chair: Ricardo Freire

Approximate and partial symmetries in Deep Learning

Speaker: Mircea Petrache

Pontificia Universidad Católica de Chile

Date: Tuesday, November 21st, 2023 at 12 Santiago time

Abstract:The image of a rotated cat still represents a cat: while this simple rule seems obvious to a human, it is not obvious to neural networks, which separately “learn” each new rotation of the same image. This applies to different groups of symmetries for images, graphs, texts, and other types of data. Implementing “equivariant” neural networks that respect symmetries, reduces the number of learned parameters, and helps improve their generalization properties outside the training set. On the other hand, in networks that “identify too much”, that is, where we impose too many symmetries, the error begins to increase, due to not respecting the data. In work with S. Trivedi, (NeurIPS 2023), we quantify this tradeoff, which allows to define the optimal amount of symmetries in learning models. I will give an introduction to classical learning theory bounds, and our extension of the ideas to the study of “partial/approximate equivariance”. In passing, I’ll describe some possible directions for working with partial symmetries in specific tasks.

Venue:  Sala John Von Neumann, 7th floor, Beauchef 851 / Online via Zoom
Chair: Ricardo Freire

Desde las EDP a la educación

Speaker: Patricio Felmer

DIM – CMM, Universidad de Chile

Date: Monday, November 6th, 2023 at 12 Santiago time

Abstract: El título de este seminario tiene varias lecturas: La primera es literal en el contexto de este seminario “Desde las Ecuaciones Diferenciales Parciales a la educación” lo que no parece tener mucho sentido, la segunda “Desde las Estrategias de Desarrollo Profesional a la educación escolar” sí tiene sentido y se relaciona con lo que actualmente hago en el ámbito de la investigación y principalmente en el ámbito del desarrollo. La tercera es la lectura que deja EDPa en forma ambigua y se refiere a lo que intentaré hacer en mi presentación: contar mi experiencia en la transición desde las EDP como área de investigación en matemática a la educación escolar como centro de mi quehacer académico, lo que complementaré presentando los principales desafíos que enfrentamos con el equipo de la Iniciativa ARPA en el ámbito del desarrollo profesional docente.

Venue:  Sala John Von Neumann, 7th floor, Beauchef 851 / Online via Zoom
Chair: Ricardo Freire

The soliton problem for water waves models with a varying medium

Speaker: María Eugenia Martínez

Université Claude Bernard Lyon 1 – Francia

Date: Monday, October 30th, 2023 at 12 Santiago time

Abstract: We focus on the study of solitary waves for two deep water wave models: the Whitham equation and the Zakharov water waves system. In particular, we analyse the behaviour of a solitary wave when it encounters a change in the environment, for example when the bottom of the domain containing the fluid is altered.

Zakharov water waves system arises as a free surface model for an irrotational and incompressible fluid under the influence of gravity. Such a fluid is considered in a domain with a rigid bottom and a free surface. In this talk, we are interested in analysing the behaviour of the solitary wave solution of the flat-bottom problem when the bottom actually presents a change at some point, where the solitary wave enters into an interaction regime with the bottom.

This is a joint work with Claudio Muñoz (Universidad de Chile) and Frédéric Rousset (Université Paris-Sud).

Venue:  Sala de Seminarios, 5th floor, Beauchef 851 / Online via Zoom
Chair: Ricardo Freire

Sphere Packings with Forbidden Distances

Speaker: Felipe Gonçalves 

Instituto de Matemática Pura e Aplicada – Brasil

Date: Monday, October 23rd, 2023 at 12 Santiago time

Abstract: We will introduce the sphere packing problem, survey past and recent developments, the breakthrough work of Viazovska, and report on recent work of mine.

The talk will be user-friendly and full of pictures.

Venue:  Sala de Seminarios, 5th floor, Beauchef 851 / Online via Zoom
Chair: Rayssa Caju

A gluing construction of singular solutions for a fully non-linear equation in conformal geometry

Speaker: Maria Fernanda Espinal Florez

PUC – Santiago, Chile

Date: Monday, October 16th, 2023 at 12 Santiago time

Abstract:

We study construction of a complete non-compact Riemannian metrics with positive constant \(σ_2\)-curvature, on the sphere \(\mathbb S^n\) with a prescribed singular set \(\Lambda\) given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than \(\frac{n−\sqrt{ n}−2}{2}\). This is a fully non-linear problem, nevertheless, we show that the classical gluing method of Mazzeo-Pacard for the scalar curvature still works in this setting since the linearized operator has good mapping properties in weighted spaces. A necessary condition is that \(2 ≤ 2k < n\).

Joint work with M. Del Mar González and Lorenzo Mazzieri.

Venue:  Sala de Seminarios, 5th floor, Beauchef 851 / Online via Zoom
Chair: Rayssa Caju

Fourier transform restriction phenomena and applications to control of dispersive equations

Speaker: Roberto Capistrano-Filho

Universidade Federal de Pernambuco – Brasil

Date: Tuesday, October 10th, 2023 at 12 Santiago time

Abstract: In 93′ Jean Bourgain discovered a subtle smoothing property of solutions of the KdV equation posed on a periodic domain. Since this celebrated article, the smoothing properties for dispersive systems are now well known. In this talk, we will see that in the last 15 years, the Bourgain spaces are fundamental to addressing the global control problems in periodic frameworks. Precisely, we will show that the propagation of compactness and regularity have been observed thanks to these spaces in various control problems for dispersive systems.

Venue:  Sala John Von Neumann, 7th floor, Beauchef 851 / Online via Zoom
Chair: Rayssa Caju

Variational techniques for non-trivial boundary conditions in fluid dynamics and beyond

Speaker: Nicolás Barnafi

CMM – Universidad de Chile

Date: Monday, October 2nd, 2023 at 12 Santiago time

Abstract: Despite the widely spread usage of no-slip boundary conditions in fluid dynamics, there exist several applications where it is not true that the fluid sticks to the boundary. Instead, the natural condition is setting the normal flux to zero, and then complementing the tangential direction with certain friction conditions known as the Navier boundary condition. In this talk, we will explore the many issues that can arise from strongly imposed boundary conditions in this setting, and how such issues can be alleviated in a way that additionally allows for extreme scale numerical simulations.

Venue:  Sala de Seminarios, 5th floor, Beauchef 851 / Online via Zoom
Chair: Rayssa Caju