*Leonid Berlyand, Department of Mathematics, Penn State University*

**PDE models of Active Matter**

In this talk we attempt to demonstrate how mathematical analysis could be helpful in the study of active matter, with the focus on active gels and cell motility.

We first discuss mathematical challenges and developments of novel mathematical tools due to out-of-equilibrium state of active matter (e.g., active cytoskeleton gels, bacterial suspensions, etc.).

Next we present three minimal PDE models of active gels: (i) phase-filed model (ii) mean curvature type free boundary model and (iii) Hele-Shaw type free boundary model. These models are designed to capture key biophysical phenomena in cell motility such as persistent & turning motion, symmetry breaking, and viscous fingering while having minimal set of parameters and variables.

Our goal is to provide theoretical understanding of cell polarity phenomenon via mathematical analysis of stability/instability and bifurcation from steady states to traveling waves. This is done by identification of key mathematical structures behind the models such as gradient coupling in Phase-Field model, Liouville type equation, Keller-Segel cross-diffusion, and nonlinearity due to the free boundary. We employ mathematical techniques of (i) sharp interface limit via asymptotic analysis, (ii) construction of steady states and traveling waves via Crandall-Rabinowitz bifurcation theory and (iii) topological methods such as Lerey-Schauder degree theory.

These are joint works with V. Rybalko (ILTPE, Kharkiv, Ukraine), J. Fuhrman (PSU & Mainz, Germany), M. Potomkin (PSU, USA).