Possible topics for postgraduate research

Geometric and stochastic analysis of interacting particle systems, optimal transport, Markov processes on metric measure spaces

The goal of this project is to reveal the structure of infinite-dimensional differential geometry behind interacting particle systems and to describe statistical physical phenomena in terms of the corresponding geometry. Roughly speaking, for a given interacting particle system of diffusion processes, a certain corresponding infinite-dimensional differential structure can be constructed on the space of configurations. It has a rich geometry structure, upon which one can build differential forms, curvatures, potential analysis etc, and this geometric structure is closely related to statistical physical properties of the corresponding particle system including the ergodicity, phase transitions of Gibss measures etc. For instance, one can characterise the ergodicity of particle systems in terms of the finiteness of the optimal transport distance (Monge-Kantrovich-Rubinstein-Wasserstein distance). Any student having a background or interest in either statistical physics, probability, differential geometry, or partial differential equations would be suitable for this project.

Contact: Kohei Suzuki.