Consider the problem of n predators X_1,...,X_n chasing a single prey X_0, all independent standard Brownian motions on the real line. If the prey starts to the right of the predators, X_k(0) < X_0(0) for all k=1,...,n, then the first capture time is
T_n = inf{ t > 0 : X_0(t) = X_k(t) for some k }. Equivalently, this is the first exit time for a Brownian motion that starts at an interior point of the corresponding cone in R^(n+1). Bramson and Griffeath (1991) showed that the expected capture time
E(T_n) = ? for n = 1, 2, 3 and, based on simulation, conjectured that E(T_n) < ? for n > 4. This conjecture was proved by W. V. Li and Q. M. Shao (2001) for n > 4 using a result of de Blassie (1987), that the finiteness of expectation is equivalent to a specific lower bound of the first Dirichlet eigenvalue of the domain which is the intersection of cone with the unit n-dimensional sphere at the origin.
I will discuss my joint work with J. Ratzkin, in which we prove the conjecture for n = 4 by establishing the eigenvalue estimate.

Special Lagrangian submanifolds are submanifolds of a Ricci-flat Kahler manifold that are both minimal and Lagrangian. We will introduce some basic facts about special Lagrangian geometry and then describe a gluing construction for special Lagrangian submanifolds.

We define a new spectrum for compact length spaces and Riemannian manifolds called the ``covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers $\delta>0$ which identify the distinct $\delta$ covers of the space. We investigate the relationship between this covering spectrum, the length spectrum, the marked length
spectrum and the Laplace spectrum. We analyze the behavior of the covering spectrum under Gromov-Hausdorff convergence and study its gap phenomenon. This is a joint work with Christina Sormani.