# Kolloquium: Marcus Waurick (TU Freiberg): Index Theorems for the Dirac Operator

Index Theorems for the Dirac Operator – Vortragender: Marcus Waurick, Technische Universität Bergakademie Freiberg – Einladender: H. Schulz-Baldes

Abstract: In mathematical physics it is a challenging

problem to compute the Fredholm index of a given partial differential operator.

For this, generally, only abstract results are available with only little

chance of providing an explicit formula for the index. In certain situations,

however, such computations are possible. In this talk we showcase these

situations for some Dirac type operators and demonstrate their fundamental

differences if considered on bounded underlying domains or on open space.

Indeed, in the former case the obtained index formula depends only on the

geometry of the underlying bounded domain, whereas in the latter case the

presented formula — also known as Callias Index Formula — solely depends on

the bounded, selfadjoint matrix-valued asymptotically unitary potential. We

demonstrate how to derive the formulas in either case and give hints as to how

the Witten index formula for the resolvents can be applied in the latter case.

The talk is based on joint work with Dirk Pauly (MZ, 301:1739-1819, 2022) and

Fritz Gesztesy (LNM, Springer, 2157, 2016).