Nate Tenpas – Vanderbilt University

We’ll show how recently introduced linear programming bounds can be used to show the optimality of certain lattice configurations in $\mathbb{R}^2$ among a large class of periodic configurations. Some of the lattice configurations address the important conjecture that the hexagonal lattice is universally optimal, while others provide a new, and largest of its kind, class of periodic energy minimizers which are not obtained from a universally optimal lattice.

Kevin Boucher – University of Southampton

Alice Mark – Vanderbilt University

The plan is to watch and discuss another of the video cases for college math instruction.

Daniel Gomez –

How long will a confined Brownian particle take to first hit a small target? It is well known that as the target size decreases the mean-first-hitting-time (FHT) diverges in dimensions two and greater, whereas it remains bounded in one dimension. What happens if the particle instead exhibits Lévy flights? In this talk I will describe how asymptotic techniques can be used to characterize the mean-FHT for a Lévy flight in a periodic one-dimensional domain. These asymptotic results show us that as the stability index of the Lévy flight is decreased we recover behavior that is qualitatively similar to that of a one-, two-, and higher-dimensional Brownian particle. This is not an accident, but rather a direct consequence of both the properties of one-dimensional Lévy flights, as well as of the singular behavior of certain fractional Green’s functions. Using this latter point as a springboard, I will conclude by outlining how similar asymptotic techniques can be used to study spike solutions to one-dimensional singularly-perturbed fractional reaction-diffusion equations.