Chern Gauss-Bonnet, Index Theory, and Fourier Transform

Chern Gauss-Bonnet, index theory, and Fourier transform In the first lecture, the hypoelliptic Laplacian was described in terms of an interpolation between operators. In the second lecture, via the Hamiltonian-Lagrangian correspondence, I will view it as realizing a version of Chern Gauss-Bonnet on the loop space of the given manifold. I will explain the relation of the hypoelliptic Laplacian to index theory, its wave like properties, and the fact that it can be understood in terms of a geometric implementation of Fourier transform.