Monday Analysis Seminar: Does diffusion determine the geometry?

2017-7-24 15:00 - 2017-7-24 16:30
Faculty of Science Building #3 Room 210
Marcel Schmidt (Mathematical Institute Friedrich-Schiller-Universität)
Can one hear the shape of a drum? In mathematical terms this famous question of M. Kac asks whether two unitarily equivalent Laplacians live on the same geometric object. It is now known, that the answer to this question is negative in general.

Following an idea of Wolfgang Arendt, we replace the unitary transformation intertwining the Laplacians by an order preserving one and then ask how much of the geometry is preserved. In this situation the associated semigroups, which encode diffusion, are equivalent up to an order isomorphism. Therefore, the question becomes as stated in the title and we try to provide an answer in great generality. In particular, we discuss the situation for graph Laplacians and Laplacians on metric measure spaces. (this is joint work with Matthias Keller, Daniel Lenz and Melchior Wirth)

Time varies from talk to talk within 14:45--18:00.

Monday Analysis Seminar