The basic idea underlying Connes' noncommutative geometry is to define a noncommutative space by its "ring of functions", which is a possibly-noncommutative ring. Geometric examples in which such spaces arise include quotient spaces by group actions, and leaf spaces of foliations. In the firstpart of the talk I will explain how one attaches noncommutative spaces to these geometric examples.
Connes proved an index theory for families of operators which are parametrized by such noncommutativespaces. His original proof used K-theory methods. In recent joint work withSasha Gorokhovsky we gave a more explicit proof of Connes' theorem, by means of heat equationmethods. I will outline the idea of the proof and discuss its applications.