Title: Inverse-spectral theory of the Koopman operator on unit phase spaces of compact Riemannian manifolds
Abstract: This project develops an inverse spectral theory for the Koopman operator on the unit phase space—modeled by the cosphere bundle of a compact Riemannian manifold $(M,g)$. Central to the theory is the flat-trace distribution $\text{Tr}^\flatV^t$, which regularizes the classical wave trace by interpreting contributions of closed geodesics as distributional singularities, thereby forming a microlocal bridge between geodesic dynamics and the spectral theory of the Laplacian. Originally introduced by Victor Guillemin in 1977, this concept is further generalized here with explicit computations on standard model cases, alongside discussions on microlocal aspects and a complex geometric framework for the flow-on-phase-space dynamics on real-analytic Riemannian surfaces.