Using the theorems we have proven for finite dimensional integrals as motivation, I will define the Euclidean correlation functions for a quantum mechanical particle moving in an arbitrary smooth potential in terms of a sum over graphs and give a derivation of the Feynman rules for this simple system.

The cross-ratio is an interesting quantity in elementary geometry because it is invariant under projective transformations. I will propose a new generalization of the cross-ratio, although showing whether the new expression gives more information than previously known invariants requires an analysis of rational functions on real and complex weighted projective spaces. This talk is based on an article appearing soon in the Journal of Mathematical Imaging and Vision, and it will be accessible to students.

This heuristic `physical proof' is due to E. Witten from 1980's and motivated the later rigorous heat equation proofs. I shall first review the Feynman path integral formalism for supersymmetric quantum mechanics. This formalism will next be applied to the Hamiltonian defined by the Dirac operator of a spin manifold, and after some non-trivial manipulations within the path integral, will lead to a proof of the index formula.I shall recall all needed background material from physics and geometry.

Following the previous talks on the Atiyah-Singer index theorem by Masoud,we will prove another important special case, namely the Hirzebruch-Riemann-Roch theorem. This theorem gives the holomorphic Euler characteristic of a holomorphic vector bundle over a compact Kähler manifold in terms of the Todd class of the manifold and the Chern character of the vector bundle. It will be shown how in the case of a holomorphic line bundle over a Riemann surface this reduces to the classical Riemann-Roch theorem.