Mathieu Lise (NYU) will give a talk on 04/08/2025.

We consider the famous Stefan problem, which the simplest mathematical model for phase transitions (e.g. the solidification of a liquid). It provides a partial differential equation with moving boundary, which is interestingly non linear but well understood in most cases. In this work we focus on one particularly non trivial case: when the liquid is “supercooled”, i.e. with initial temperature below zero. Physicians call this state “metastable” and practical observations identify blow-ups instead of a continuous evolution. The modern approach to this problem is mainly probabilistic and relies on a stochastic differential equation with a contagion term. Moreover, past research in this area has identified a striking link with the study of large particle systems that can occur in banking systemic risk or neuroscience.

This presentation sums up the work done in my master’s thesis at the University of Oxford under the supervision of Prof. Ben Hambly in 2024. It consists in a detailed literature review of the link between the supercooled Stefan problem and homogeneous interacting particle systems (especially in banking systemic risk). Finally, we will highlight a few challenges when generalizing to heterogeneous network models for interacting particle systems.