Asilya Suleymanova (Humboldt-Universität zu Berlin)
In the paper "Can one hear the shape of a drum?" Mark Kac asked the following question. If an infinite sequence of eigenmodes of a domain is given can one determine geometry of the domain? What geometrical information can we obtain? Asymptotic expansion of the heat trace is used to answer these question.
In the talk, we shall start with an introduction to the heat kernel of the Laplace operator on a manifold. In the case of a compact smooth manifold, heat trace expansion gives some geometrical information such as dimension, volume and total scalar curvature of the manifold. Next we shall consider a space with conical singularities. What information can one derive from the heat trace expansion on it? Can one hear the presence of a singularity? I present results in dimension 2 and 4.
