When: Sep. 26th 2019 (Thurs), 16:00 -- 17:15.
Where: Building 110, Room N. 103.

Title: The heat kernel asymptotics and the zeta-determinants of elliptic operators

Abstract: The heat kernel asymptotics of a Laplacian on a compact Riemannian manifold contains many geometric informations. Indeed, each coefficient is expressed by some curvature tensors, for example, Riemannian curvature tensor, Ricci tensor, scalar curvature and principal curvatures, etc, from which one can extract some geometric informations.
On the other hand, the zeta-determinants of a Laplacian is a global spectral invariant which play a central role in the analytic torsion. It is a direct generalization of the determinant of a linear map acting on a finite dimensional vector space, and is introduced by Ray and Singer in 1970’s to define the analytic torsion, which is an analytic counterpart of the Reidemeister torsion. In this talk, I’m going to discuss some basic facts of the zeta-determinants, its gluing formula, the heat kernel asymptotics and the Atiyah-Singer Index Theorem. The gluing formula of a zeta-determinant contains the so-called Dirichlet-to-Neumann operator, which is similar to the operator in the Steklov eigenvalue problem. I’m going to show that each coefficient in the asymptotic expansion of the zeta-determinant of a one parameter family of the Dirichlet-to-Neumann operators is expressed by some curvature tensors, which is analogous to the heat kernel asymptotics of a Laplacian.