The higher dimensional Beltrami equations and its connections to Hodge theory

I will show how one can solve a second order scalar elliptic equation using Hodge theory. This will lead to a generalization of the classical Beltrami equation in the plane to higher dimensions using Dirac operators. It will also serve as an excuse to discuss boundary value problems for elliptic Dirac operators and to introduce the notion of quasiconformal structures on topological manifolds due to Dennis Sullivan. This permits one to do analysis on manifolds without any smooth structure. If time permits, I will also indicate how this is used in the proof of the index theorem of Connes, Donaldson, Sullivan and Teleman on topological mainfolds.