Proper Actions and Index Theory of Elliptic Operators

Hang Wang  (University of Adelaide)

10:00- 11:00, May 16, 2017   Science Building A1510

Abstract:

In differential geometry, Atiyah-Singer index theorem, by means of calculating the Fredholm index of an elliptic operator on a compact manifold, creates an amazing link between geometric data and topological invariants of the manifold. It unifies in various geometric settings several beautiful theorems, such as the Gauss-Bonnet theorem and the Riemann-Roch theorem. One of the most exciting generalisations of the Atiyah-Singer index theorem is index theory of elliptic operators on a noncompact manifold with proper group actions, due to its connection to several important subjects and problems in topology, geometry and representation theory. For example, a famous open problem in topology about homotopy invariance of higher signatures, known as the Novikov Conjecture, is formulated in this framework. After the introduction, we will present an application of index theory of invariant elliptic operators in representation theory, where a geometric method can be used to recover Harish-Chandra’s character formula for discrete series representations of a semisimple Lie group. This is joint work with Peter Hochs.

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