Mathematics

A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. A minimal two species periodic assembly is one with the least energy per lattice cell area. There is a parameter b in [0,1] and the type of the lattice associated with a minimal assembly varies depending on b. There are several thresholds defined by a number B=0.1867... If b is in [0, B), a minimal assembly is associated with a rectangular lattice; if b is in [B, 1-B], a minimal assembly is associated with a square lattice; if b is in (1-B, 1], a minimal assembly is associated with a rhombic lattice. Only when b=1, this rhombic lattice is a hexagonal lattice. None of the other values of b yields a hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where hexagonal lattices are ubiquitously observed.

The interplay between 1D traveling pulses with oscillatory tails (TPO) and heterogeneities of bump type is studied for a generalized three-component FitzHugh-Nagumo equation. We first present that stationary pulses with oscillatory tails (SPO) forms a “snaky" structure in homogeneous space, then TPO branches take a form of "figure-eight-like stack of isolas" located close to the snaky structure of SPO. Here we adopt voltage-difference as a bifurcation parameter. A drift bifurcation from SPO to TPO can be found by introducing another parameter at which these two solution sheets merge. As for the heterogeneous problem, in contrast to monotone tail case, there appears a nonlocal interaction between the TPO and the heterogeneity that creates infinitely many saddle solutions. The response of TPO shows a variety of dynamics including pinning and depinning processes in addition to penetration and rebound. Stable/unstable manifolds of these saddles interact with TPO in a complex way, which causes a subtle dependence on the initial condition and a difficulty to predict the behavior after collision even in one-dimensional space. Nevertheless, for 1D case, a systematic global exploration of solution branches (HIOP) induced by heterogeneities, and the reduction method to finite-dimensional ODEs allow us to clarify such a subtle dependence of initial condition and detailed mechanism of the transitions from penetration to pinning and pinning to rebound from dynamical system view point. It turns out that the basin boundary between two different outputs against the heterogeneities forms an infinitely many successive reconnections of heteroclinic orbits among those saddles as the height of the bump is changed, which causes the subtle dependence of initial condition. This is a joint work with Takeshi Watanabe.

Cell migration plays a central roles in embryonic development, wound healing and immune surveillance. In 2008, Yoichiro Mori, Alexandra Jilkine and LEK published a reaction-diffusion model for the initial step of cell migration, the front-back chemical polarization that sets a cell's directionality. (More detailed mathematical properties of this model were described by the same group in 2011.) Since then, progress has been made in investigating how that simple "wave-pinning" mechanism is shaped and tuned by feedback from other proteins, from the cell's environment (extracellular matrix), from interplay with larger signaling networks, and from cell-cell interactions. In this talk, we will describe some of this progress and mathematical questions that arise. In particular, AB will demonstrate how his numerical PDE bifurcation analysis has helped us to understand how cells repolarize to reverse their direction of motion.

Any countable group G gives rise to a von Neumann algebra L(G). The classification of these group von Neumann algebras is a central theme in operator algebras. I will survey recent rigidity results which provide instances when various algebraic properties of groups, such as the presence or absence of a direct product decomposition, are remembered by their von Neumann algebras. I will also explain the strongest such rigidity results, where L(G) completely remembers G, and discuss some of the open problems in the area.

Mirzakhani gave an inductive procedure to build random hyperbolic surfaces by gluing together smaller random pieces along curves. She proved that as the length of the gluing curve grows, these families equidistribute in the moduli space of hyperbolic surfaces. In this talk, I’ll explain how the conjugacy (exposited in James’s talk) between the earthquake and horocycle flows provides a template for translating equidistribution results for flat surfaces into equidistribution results for hyperbolic ones. Using this correspondence, we address Mirzakhani’s twist torus conjecture and exhibit new limiting distributions for hyperbolic surfaces built out of symmetric pieces. This is joint work (in progress) with James Farre.