In the past decade, there has been extensive research on the nonlinear Schrödinger equation (NLS) on metric graphs, driven by both the physical and mathematical communities. Metric graphs, in essence, are one-dimensional objects that can model network-like structures. The first goal of this talk, particularly for those who are new to metric graphs, is to provide an introduction to these structures and present the appropriate functional framework for studying the NLS equation on them.

It is well-established that both the metric (size) and topological (shape) properties of the graphs can impact the existence of solutions to the NLS. As a result, no general theory currently exists for analyzing the NLS equation on metric graphs. A common approach is to focus on specific classes of graphs. In this talk, we focus on two such graphs: the $\mathcal{T}$-graph and tadpole graphs. We then discuss, using techniques from the theory of ordinary differential equations (specifically, parts of the period function), how to approach questions related to the existence, uniqueness, and multiplicity of positive solutions on these graphs.

Time permitting, we will demonstrate how this careful analysis leads to a series of existence and uniqueness/multiplicity results for a class of graphs known as single-knot graphs.

This talk is based on joint work with Simão Correia and Hugo Tavares.

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 the development of new mathematical tools. We focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction. We introduce a minimal two-dimensional free-boundary PDE model that captures the evolution of the cell shape and nonlinear diffusion of myosin.

We first consider a linear diffusion model with two sources of nonlinearity: Keller-Segel cross-diffusion term and the free boundary that models moving/deformable cell membranes. Here we establish asymptotic linear stability and derive the explicit formula for the stability-determining eigenvalue.

Next, we consider the effect of nonlinear myosin diffusion, which results in the change of the bifurcation type from super- to subcritical, and we obtain an asymptotic representation of the bifurcation curve (for small velocities). This allows us to derive an explicit formula for the curvature at the bifurcation point that controls the bifurcation type. In the most recent work in progress with the Heidelberg biophysics group, we study the relation between various types of nonlinear diffusion and bistability.

Finally, we discuss novel mathematical features of this free boundary model with a focus on non-self-adjointness, which plays a key role in the spectral stability analysis. Our mathematics reveals the physical origins of the non-self-adjoint of the operators in this free boundary model.

Joint works with A. Safsten & V. Rybalko (Transactions of AMS 2023, and Phys. Rev. E 2022), with O. Krupchytskyi &T. Laux (Preprint 2024), and with A. Safsten & L. Truskinovsky ( Arxiv preprint 2024). This work has been supported by NSF grants DMS-2404546, DMS-2005262, and DMS-2404546.