Hide-and-Seek and Spectral Invariants (on Markov Chains and Surfaces)

In this talk, I will describe a Hide--and--Seek game that gives a probabilistic interpretation of two trace-like spectral invariants. I will begin by describing the Markov Chain setting, in which a classical spectral invariant known as Kemeny's constant arises. I will then discuss (a suitably modified) game that a Brownian particle might play on a surface: here, the spectral invariant is a regularized 'trace' for the Laplacian operator, which contains geometrical information.