We continue on: ---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory, ---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch.

Noncommutative geometry was discovered by Alain Connes in the early 80's. In short, noncommutative geometry is based on a duality between the categories of algebra and geometry. A noncommutative space is thought of as the dual object to a noncommutative algebra. In my talk, I will discuss the ideas of noncommutative geometry and concepts in the category of geometry which admit noncommutative versions, and of course examples.