## Noncommutative Geometry Seminar

**Date:** November 20, 2019

**Time:** 2:00PM - 3:00PM

**Location:** BLOC 628

**Speaker:** Peter Hochs, University of Adelaide

**Title:** *A localised equivariant index for proper actions and an APS index theorem.*

**Abstract:** Roe defined a localised version of the coarse index of an elliptic operator that is invertible outside a subset Z of the manifold M it is defined on. An equivariant version of this index was defined for proper and free actions by discrete groups by Xie and Yu. With Guo and Mathai, we extended this to proper actions by any locally compact group G. If Z/G is compact, then this index takes values in the K-theory of the group C* algebra of G, and generalises the Baum-Connes analytic assembly map. It also generalises an equivariant index of Callias-type operators constructed earlier by Guo. Another special case is an equivariant index for proper, cocompact actions on manifolds with boundary, generalising the Atiyah-Patodi-Singer (APS) index and its equivariant version. With Bai-Ling Wang and Hang Wang, we obtained an equivariant APS index theorem in this context. Using a version for maximal group C*-algebras and Roe algebras, we obtain a link with an index on invariant sections defined earlier with Mathai.