Abstract: In many geometric situations we may encounter the action of a group G on a manifold M, e.g., in the context of foliations. If the action is free and proper, then the quotient M/G is a smooth manifold. However, in general the quotient M/G need not even be Hausdorff. Under these conditions how can we do diffeomorphism-invariant geometry?

Noncommutative geometry provides us with a solution by trading the badly behaved space M/G for a non-commutative algebra, which essentially plays the role of the algebra of smooth functions on that space. The local index formula of Atiyah and Singer ultimately holds in the setting of noncommutative geometry. This enabled Connes and Moscovici to reformulation of the local index formula in the setting of diffeomorphism-invariant geometry.

The first part of the lectures will be a review of noncommutative geometry and Connes-Moscovici's index theorem in diffeomorphism-invariant geometry.

In the 2nd part, I will hint to on-going projects on the reformulation of the local index formula in two new geometric settings: biholomorphism-invariant geometry of strictly pseudo-convex domains and contactomorphism-invariant geometry of contact manifolds.

Abstract: We define the Lebesgue integral directly by extending the Riemann integral. We will see that the passage of limit under the integral sign can be used effectively to define the Lebesgue integral, and avoids all the usual definitions of measures and measurable sets. This talk is also available to all undergrad students who have good backgroud in calculus.