Mathematics Colloquia and Seminars

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Suppose n pursuers starting at the origin chase a single prey starting at 1, all doing standard independent Brownian motions on the real line. Bramson and Griffeath (1991) showed that the expected capture time is infinite for three or fewer pursuers and, after simulations, conjectured that it is finite for four or more. Li and Shao (2001) proved it for five or more pursuers and in recent work with Ratzkin, we show that it finite for four, completing the proof. We use the idea of Li and Shao to reduce the problem to an estimate of the first Dirichlet eigenvalue of a domain in the sphere.

I'll discuss eigenvalues, describe the reduction, the eigenvalue estimates, and some related numerics.