16 March – Matthias Keller (Universit├Ąt Jena)

“Can one hear the shape of a drum?” is a famous question by Mark Kac. This question asks whether the spectrum of the Laplacian determines the underlying domain. However, it is well known that the answer is no in general for domains as well as for graphs. Abstractly the question of Kac is whether a unitary transformation of the Laplacians preserves the geometry. Following ideas of Wolfgang Arendt, we reformulate the question by replacing the unitary transformation by an order isomorphism. This leads to the question whether a transformation of the heat semigroups under order isomorphisms determines the geometry. We address this question for general weighted graphs. (This is joint work with Daniel Lenz, Marcel Schmidt and Melchior Wirth)