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).