There are some natural situations in which self-adjoint operators cannot exhibit spectral gaps for topological reasons. Perhaps the most prominent examples come from the theory of topological insulators, where boundaries very generally force the appearance of spectrum inside bulk gaps. Closely related phenomena are spectral flows, where topology can stabilize gap closings of continuous families of self-adjoint operators, leading for example to robust eigenvalue crossings. In many cases, these effects can be understood in a unified way using K-theory for C*-algebras. In this talk, I want to explain the basic mechanism behind such topological gap-filling and illustrate it through some examples.

Fixed measure scaling limits of Christoffel--Darboux kernels encode information about orthogonal polynomials, such as the local distribution of their zeros. Different limit kernels are associated with different universality classes, e.g. sine kernel with bulk universality and locally uniform zero spacing. We will describe necessary and sufficient conditions for a class of scaling limits corresponding to homogeneous de Branges spaces; this includes bulk universality, hard edge universality, and other notable classes. The talk is based on joint work with Benjamin Eichinger and Harald Woracek.

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.