Introduction to Index Theory（5 Lectures）

Hang Wang 王航  (ECNU)

13:00-14:15, Oct 1-5, 2018   Science Building A510

Abstract:

I will talk about the K-theoretic index index theory of an elliptic operator on a manifold invariant under the action of a group (as appeared in the setting of the Baum-Connes assembly map) and some concrete ways of getting cohomological formulas by taking traces on the index in K-theory. These formulas are generalizations of the Atiyah-Singer index formula and are motivations for constructing topological invariants for the space and for its group of symmetries. Outline: 1. Fredholm index in K-theory, Dirac operators and heat kernel method; 2. L^2-index theorems (and applications) by Atiyah and by Connes-Moscovici; 3. K-theoretic index for equivariant elliptic operators, example of Connes-Kasparov’s isomorphism, relation to L^2-index; 4. Pairing of K-theoretic index with a trace associated to a conjugacy class of the group, applications.

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