Nonlinear Diffusion and nonlocal Interaction Models - Entropies, Complexity, and Multi-Scale Structures (23w6003)


(University of L'Aquila)

(Universidad Autónoma de Madrid and ICMAT)

Katy Craig (University of California, Santa Barbara)



The Institute of Mathematics at the University of Granada will host the "Nonlinear Diffusion and nonlocal Interaction Models - Entropies, Complexity, and Multi-Scale Structures" workshop at the University of Granada (IMAG) in Spain, from May 28 - June 2, 2023.

Mathematical models with nonlinear diffusion and many-body systems have a long history in applied mathematics, dating back to classical applications in physics. Their tremendous flexibility to describe unrelated phenomena has made them omnipresent across the sciences: from physic, biology, economics, social sciences, to artificial intelligence, machine learning, data science and complex systems. Similarly, their mathematical study spans several fields within mathematics, including local and nonlocal partial differential equations of nonlinear diffusion type, many-particle systems with interactions, kinetic equations, network and graph based modelling. While these fields feature each one a somehow independent literature, it has become clear in recent years that a deep connection among them is present.

Well established domains of mathematical analysis such as functional inequalities, gradient flows, mean-field limits and entropy methods are the main examples of common fields of interplay for those families of models. Outstanding results have been recently obtained in the literature with respect to well-posedness, stability for sharp inequalities, asymptotic behavior, and mean-field limits, and yet so many crucial questions remain open, requiring the development of more sophisticated analytical tools. In particular, the connection of nonlinear diffusions and nonlocal many-particle systems with artificial intelligence and machine learning came up quite recently and is a very fertile ground for cross sharing of new ideas and innovative, interdisciplinary thinking.

The program's primary objective is to foster new collaborations, both on existing links within the analysis of PDEs and calculus of variations communities, and towards a closer interplay with machine learning and artificial intelligence. We will foster the communication of the latest vanguard results of interest for the above fields, to find a common ground of collaboration and promote cross-pollination of the technical advancements. To this aim, the program will feature both world leading experts in nonlinear diffusion models, nonlocal many-particle systems modelling, kinetic PDEs, and machine learning modelling, together with \normalcolor some of the most promising young researchers in those fields.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. BIRS is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).