Abstract: I will present two results on the asymptotic stability of small solitary waves for the one-dimensional cubic-quintic Schrödinger equation. The first result concerns the focusing-defocusing double power nonlinearity, for which the linearized operator around the small solitary waves has no internal mode. The second result concerns the more delicate case of the focusing-focusing double power nonlinearity, for which the linearized operator around the small solitary waves actually has an internal mode. The internal mode component of the solution is controlled by checking explicitly a condition related to the Fermi golden rule. We will also explain the analogies and differences with previous similar works, mainly on nonlinear wave-type models with solitons or kinks.

References : Yvan Martel, Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode, arXiv:2110.01492
Yvan Martel, Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation, arXiv:2312.11016

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

Chair: Ricardo Freire

Abstract: We introduce and study a new model for the progression of Alzheimer’s disease incorporating the interactions of A_beta-monomers, oligomers, microglial cells and interleukins with neurons through different mechanisms such as protein polymerization, inflammation processes and neural stress reactions. In order to understand the complete interactions between these elements, we study a spatially-homogeneous simplified model that allows to determine the effect of key parameters such as degradation rates in the asymptotic behavior of the system and the stability of equilibriums. We observe that inflammation appears to be a crucial factor in the initiation and progression of Alzheimer’s disease through a phenomenon of hysteresis, which means that there exists a critical threshold of initial concentration of interleukins that determines if the disease persists or not in the long term. These results give perspectives on possible anti-inflammatory treatments that could be applied to mitigate the progression of Alzheimer’s disease. We also present numerical simulations that allow to observe the effect of initial inflammation and concentration of monomers in our model.

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

Chair: Ricardo Freire

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

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

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.

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

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

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

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.

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

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.

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

Abstract:

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