Sergey Nadtochiy (IIT Math) - Mean Field Systems for solving Free–Boundary problems: From Finance to Neuroscience and Physics

Description:This talk is concerned with the recent advances in the study of particle systems with threshold-type interaction. Such systems appear naturally in the mathematical models for neuron cells, supercooled liquids, computer and credit networks, etc. In particular, our recent work shows how to use these particle systems in order to solve the supercooled Stefan PDE. The latter is a free-boundary problem which describes the process of freezing of a supercooled liquid, and whose well-posedness has been an open question for the last 50 years. We prove the well-posedness and provide a complete description of such solutions in one space dimension, by relating the temperature distribution in the liquid to the regularity of the interface between the solid and liquid states. These results provide a methodology for establishing the well-posedness and describing the structure of other mean-filed systems with threshold-type interaction (and of the associated free-boundary problems). This is a joint work with F. Delarue and M. Shkolnikov.