Abstract: In this lecture, we continue with the following topics: ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weitzenbock formula. ---Heat kernel and its asymptotic expansion, Gilkey's formula, McKean-Singer formula.

Abstract: I will discuss the following result, which is joint work with A. Sukhov: on any compact real surface $S$, except the sphere and $RP^2$, there exist two smooth functions with the property that their rational combinations are dense in the space of continuous functions on $S$. The proof relies on many different results in several complex variables, symplectic geometry, and approximation theory. I will give a general overview of this.