Title: Steklov Spectral Invariants via Guillemin-Wodzicki Residue and Applications in Inverse Spectral Problem

Speaker: 员含章（University of Washingtong at St. Louis)

Time: 5.13.2025, 16:00-17:30

Place: Second Teaching Building, 2303 

Abstract: Given a pseudodifferential operator, its Guillemin-Wodzicki residue is a global volume density defined in terms of the symbol expansion in local coordinate systems. By computing the Guillemin-Wodzicki residue of Dirichlet-to-Neumann operators, we obtain two new Steklov spectral invariants for compact $3$-dimensional Riemannian manifolds with smooth boundary. As an application, we show that every geodesic ball in the simply connected $3$-dimensional space form with given curvature $\kappa$ can be determined by its Steklov spectrum among all compact Riemannian manifolds with constant curvature $\kappa$ and smooth boundary if $\kappa\le 0$. If $\kappa > 0$, we prove the same result under slightly extra assumptions.