Abstract: Motility-induced phase separation
Best known by its acronym---MIPS---motility-induced phase separation is a fascinating phenomenon encountered in active matter systems---such as bacterial colonies. In certain conditions, an initially homogeneous colony spontaneously divides into regions of different densities. This and related phenomena are an area of active research in physics of active matter---experimental, theoretical and numerical. I will propose a model which allows to derive some of the key features of MIPS. This model leads to a system of stochastic differential equations, and, further, to a study of stability of solutions of a related partial differential equation. The subject needs both rigorous mathematical work and numerical simulations. Several natural problems can be formulated that would advance the understanding of MIPS, including projects for interested graduate students.