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.

Projected Green’s Functions (pGf), Gxx(⍵), have long been used to describe the localization of quantum systems. More recently, pGf zeros have been used to determine physical observables of topological invariants in free-fermion systems, including topological obstructions to bulk localization and bulk-boundary correspondence. In this talk, I will discuss how these pGfs appear in transfer matrices and what their zeros can tell us about the solutions to transfer matrix equations – linking the localization and topological perspectives. Using these methods, we re-examine the almost-Matthieu operator and notice new guarantees on analytic regions of its resolvent.

We take a refreshing new look at boundedly rational quadratic models in economics using some elementary modeling of the principles put forward in the book Humanomics by Vernon L. Smith and Bart J. Wilson. A simple model is introduced built on the fundamental Humanomics principles of gratitude/resentment felt and the corresponding action responses of reward /punishment in the form of higher/lower payoff transfers. There are two timescales: one for strictly self-interested action, as in economic equilibrium, and another governed by feelings of gratitude/resentment. One of three timescale scenarios is investigated: one where gratitude /resentment changes much more slowly than economic equilibrium (“quenched model”). Another model, in which economic equilibrium occurs over a much slower time than gratitude /resentment evolution (“annealed” model) is set up, but not investigated. The quenched model with homogeneous interactions turns out to be a non-frustrated spin-glass model.  For this particular model, the Nash equilibrium has no predictive power of Humanomics properties since the rewards are the same for self-interested behavior, resentful behavior, and gratitude behavior. Accordingly, we see that the boundedly rational Gibbs equilibrium does indeed lead to richer properties.

we study 1-d random Schrödinger operators on a finite interval with Dirichlet boundary conditions. We are interested in the approximation of the ground state energy using the minimum of the effective potential. For the 1-d continuous Anderson Bernoulli model, we show that the ratio of the ground state energy and the minimum of the effective potential approaches 

π^2/8