## Geometry Seminar - Abstracts

### Talk

Thursday 25 October 2018, 16:00-17:00 in HG03.085

**Peter Hochs** (Adelaide)

*Orbital integrals in index theory and K-theory*

### Abstract

An orbital integral of a function on a group \(G\) is its integral
over a conjugacy class in \(G\). If such an orbital integral defines a
continuous functional on a convolution algebra \(A(G)\) of
functions on \(G\), then it is a trace on that algebra. If the
conjugacy class consists of just the identity element, this is the
classical von Neumann trace. In general, such a trace induces a
map on the \(K\)-theory of \(A(G)\)with values in the complex
numbers. If \(A(G)\) is dense in the reduced or full group
\(C^*\)-algebra of G and closed under holomorphic functional
calculus, then this gives a map on the K-theory of that group
\(C^*\)-algebra. It has turned out in recent years that such maps
are useful tools for studying elements of these \(K\)-theory
groups. This is true in particular for \(K\)-theoretic indices of
\(G\)-equivariant elliptic operators. Index formulas for the
numbers obtained in this way have turned out to have implications
to representation theory and geometry. In this talk, I will
discuss this development, including joint work with Hang Wang.

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