Marcel Schmidt - Does diffusion determine the geometry?, 02nd July 2014

A famous question of M. Kac asks: Can one hear the shape of a drum? In mathematical terms he asked whether the spectrum of the Laplacian uniquely determines the geometry of the underlying domain or, put differently, whether two unitary equivalent Laplacians live on the same geometric object (up to isometry). It is now known, that the answer to this question is no, in general. Following an idea of Wolfgang Arendt, we replace the unitary transformation intertwining the Laplacians by an order preserving one and then ask the same question. As in this situation the associated semigroups are equivalent up to an order isomorphism our question becomes as stated in the title. In this talk we will give an overview over the recent results concerning this problem. In particular, we will discuss which quantities are determined by diffusion and give some possible applications. (this is joint work with Matthias Keller, Daniel Lenz and Melchior Wirth)