> CUWB-III: 31.01.24 - 02.02.24

Differential and difference equations in models of physics and biology

Venue: Unidad Academica de Sistemas Arrecifales, Puerto Morelos, Quintana Roo, Mexico

The objective of the workshop is to explore the possibilities for collaboration between the CUWB mathematical community and the staff of the Institute of Marine Sciences and Limnology. It is planned that researchers of the Institute interact with mathematicians so that open questions in marine sciences can be formulated in mathematical language so that they can be addressed in multidisciplinary projects.
 

Workshop lectures:
 

Miguel Ballesteros (IIMAS-UNAM)
Title: A fresh perspective on jungles: a model for species distribution inspired by quantum physics

Abstract: A journey that begins with expeditions to the jungle (and cloud forests) to collect data, observing individual amphibians. We delve into the world of physics, identifying individuals as particles. We use physics to formulate the model intuitively and mathematics to implement it precisely. We apply computational algorithms.

Kirill Cherednichenko (University of Bath)
Title: Bridging scales by asymptotics 

Abstract: I will show how dimensional analysis of continuous media naturally prompts the use of asymptotic methods to derive simplified "effective" models from complex ones. This provides a mathematical framework for the link between mathematical models of physical phenomena at different length-scales and serves as a helpful tool to set up efficient numerical procedures.  
 

Fedro Guillén  (IIMAS-UNAM)
Title: A new population model inspired by statistical physics, quantum mechanics, and artificial intelligence

Abstract: In this talk, a novel method for estimating the population of certain amphibian families in a delimited geographical region will be presented. Specifically, a spatial grid is set over the region, and in each plot, a random variable is considered to count the number of specimens within it. Assuming that the variables only depend on their first-order neighbours, the set of random variables forms a Markov random field whose distribution is, in several cases, a Gibbs measure, as demonstrated in the Hammersley–Clifford theorem. Building on this, a Gibbs measure is associated with the variables, where the energy function can be decomposed into the sum of a kinetic term that drives frogs to move randomly and a potential term that attracts or repels amphibians in each plot. By framing the energy function in this way, it can be represented as the quadratic form associated with a Schrödinger-type operator on the graph, allowing the problem to be reformulated in the context of quantum physics, as we need to find the minimum energy state associated with this operator. Therefore, the problem is reduced to modelling the potential term. This is accomplished using artificial intelligence and deep learning techniques to fit a function that depends on certain geographic variables using real data obtained in field surveys.

Armando Martínez-Pérez (IIMAS-UNAM)
Title: Analysis of the momentum operator associated with a quantum
particle in a box

Abstract: In this talk, we consider the mathematical model of the momentum concept in the non-relativistic quantum mechanics for a particle in an infinite well. We show and discuss unitarily equivalent operators associated with the momentum observable in the coordinate, momentum, and Paley-Wiener spaces. Finally, we contrast these unitarily equivalent operators with the multiplication operator in momentum space, which has raised some doubts about the definition of the momentum operator and its expectation values.

Iván Naumkin (IIMAS-UNAM)
Title: Wave evolution in a system of Schrödinger equations with exceptional potentials

Abstract: In this talk, we consider the nonlinear matrix Schrödinger equation on the half-line with general self-adjoint boundary conditions with an external potential that can be either generic or exceptional. This kind of systems corresponds to a star graph describing the behavior of n connected very thin quantum wires that form a graph with only one vertex and a finite number of edges of infinite length. We consider the scattering problem for this model. We show that in the scatteringsupercritical regime, the small solutions scatter to the corresponding linear solutions. Moreover, thanks to the general boundary conditions, we obtain a scattering result for a system of nonlinear Schrödinger equations on the line with a potential and point interactions.

Sergio Palafox (Universidad Tecnológica de la Mixteca)
​Title: Spectral analysis for a mathematical model of a mechanical system of interacting particles

Abstract: Jacobi operators are a particular realization of a certain class of symmetric operators that appear in a wide variety of mathematical models in physics and biology. This talk presents a generalization of these operators from a general perspective, starting from the mathematical model of the small oscillations of a system of interacting particles. Recent developments in spectral theory are used to tackle the direct and inverse spectral problem for this class of operators.

Luis O Silva (IIMAS-UNAM)
Title: Mathematical modelling

Abstract: This talk introduces the basics of mathematical modelling in a broad sense and presents the archetypical mathematical models of natural and social phenomena. The particularities of different kinds of mathematical models are explored from the viewpoint of mathematics and in the relationship between the theory and the modelled phenomena. In doing so, we touch upon the actual meaning of various mathematical models in science and engineering. 
 

Pre-recorded lectures:

Gerardo Franco (IIMAS-UNAM)  

Title: Teoría de dispersión y Teorema de Levinson para pperadores matriciales de Schrödinger en la línea discreta

Abstract: Consideramos operadores que son perturbaciones de primer momento del operador laplaciano discreto en la línea. Desarrollamos la Teoría de Dispersión en este contexto y derivamos fórmulas explícitas para la matriz de dispersión en términos de soluciones particulares de la ecuación discreta de Schrödinger. Además, extendemos estas fórmulas a los umbrales del espectro del Operador. Finalmente, obtenemos una relación entre el cambio de fase de la matriz de dispersión con el espectro discreto y los estados semi-acotados del operador de Schrödinger.

Diego Iniesta (IIMAS-UNAM) 

Title: Existencia de resonancias en un modelo de la electrodinámica cuántica no relativista

Abstract: En los últimos años, se han investigado las propiedades espectrales de átomos no relativistas mínimamente acoplados al campo electromagnético cuantizado. Se han demostrado resultados como la existencia de estados fundamentales y resonancias en el caso de acoplamiento singular utilizando la técnica del grupo de renormalización espectral. En esta presentación, ofrecemos una prueba alternativa de la existencia de resonancias para el modelo de Pauli-Fierz aplicando un análisis de escalas múltiples.

Enrique Álvarez del Castillo (IIMAS-UNAM) 

Title: Un criterio para la inestabilidad no lineal de soluciones de tipo onda viajera periódica para leyes viscosas de balance

Abstract: Las ecuaciones conocidas como leyes viscosas de balance poseen dos familias distintas de soluciones de tipo onda viajera periódica. La primera consiste en ondas periódicas de amplitud pequeña con periodo fundamental finito que emergen de una bifurcación (local) de Hopf alrededor de un valor crítico de la velocidad. La segunda familia está compuesta por ondas cuyo periodo tiende a ser arbitrariamente grande al surgir de una bifurcación (global) homoclínica y al aproximarse a un pulso viajero conforme su periodo crece a infinito. Se prueba que, al considerar la linealización alrededor de éstas, el espectro (continuo) de Floquet resultante intersecta el semiplano izquierdo inestable de valores complejos con parte real positiva, una condición conocida como inestabilidad espectral.

Posteriormente, mediante el buen planteamiento de las ecuaciones en cuestión en espacios de distribuciones periódicas, se demuestra que la inestabilidad espectral exhibida por los elementos de ambas familias garantiza su inestabilidad orbital en un espacio de Sobolev adecuado. La utilidad de este criterio de inestabilidad no lineal se ilustra mediante la aplicación del mismo a los elementos de las dos familias de ondas periódicas, garantizando así su inestabilidad orbital en un espacio de Sobolev periódico cuyo periodo coincide con el periodo fundamental de las ondas en cuestión.

Jonas Schober (IIMAS-UNAM) 

Title: One-parameter semigroups of unitary endomorphisms of standard subspaces and reflection positivity

Abstract: Motivated by the Haag-Kastler theory of local observables in Quantum Field Theory, one is interested in unitary endomorphisms of standard subspaces. This talk focuses on one-parameter semigroups of these unitary endomorphisms and shows how they relate to reflection positivity, Hankel operators, and Pick functions.