Explore sharp convergence rates in Wasserstein distance for empirical measures of diffusion processes on Riemannian manifolds in this 47-minute mathematical lecture. Discover how renormalization limits are explicitly formulated using the volume of the manifold along with eigenvalues and eigenfunctions of the associated elliptic operator. Gain insights into the spectral representations that characterize these Wasserstein limits, providing a deep understanding of the geometric and analytical structures underlying empirical measure convergence on curved spaces. Learn about the interplay between differential geometry, probability theory, and spectral analysis as applied to diffusion processes on manifolds.