We begin with a brief overview of the rapidly developing research area of active matter (a.k.a active materials). These materials are intrinsically out of  equilibrium resulting in novel physical properties whose modeling

requires development of new mathematical tools.  We next focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on

a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two-dimensional

free-boundary PDE  model that generalizes a previous one-dimensional model by combining a Keller-Segel model,  Hele-Shaw kinematic boundary condition, and the Young-Laplace law with a  novel nonlocal  regularizing term.   This nonlocal term precludes blowup or collapse of the cell by ensuring that membrane-cortex interaction is sufficiently strong. We found a family of asymmetric traveling  wave solutions bifurcating from stationary solutions. Our main result is the nonlinear asymptotic stability of traveling wave solutions that model observable steady cell motion. We derived and rigorously justified an explicit asymptotic formula for the stability determining eigenvalue via asymptotic expansions in a small speed of cell. Our spectral analysis reveals the physical mechanisms of stability/instability.  It also leads to a novel spectral properties  due to the non-self-adjointness of the linearized  problem which is a signature  of active matter out-of-equilibrium systems.  This results in striking math features such as collapse of eigenspaces and presence of generalized  eigenvalues and  we determine their physical origins.

This is joint work with  V. Rybalko and C. Safsten published in Transactions of AMS (2023)   and Phys. Rev.B, 2022.

If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading.