Extensions of index theory inspired by scattering theory (17frg668)

Arriving in Banff, Alberta Sunday, June 18 and departing Sunday June 25, 2017


(Australian National University)

(Baylor University)


The main aim of this focussed research group proposal is to investigate extensions of index theory to situations where one starts a priori with non-Fredholm operators. This entails generalisations of earlier work of Gesztesy et al. on this problem for particular non-Fredholm operators. We are inspired by the trace formula of Carey et al. and recent progress in scattering theory through investigations of the spectral shift function. A key question is whether the spectral shift function can be really thought of as defining a generalisation of spectral flow to non-Fredholm situations. As an adjunct to this, we will investigate the Witten index as a method of calculating this generalised spectral flow via a far-reaching generalisation of both Pushnitski's ideas on spectral shift functions and all of the previous research on this topic by the proposed participants. We will also study applications to spectral geometry and problems in condensed matter physics involving magnetic Dirac operators and their perturbations as arise for example in the study of graphene and topological phases of matter. We aim to resolve questions raised by theorists in these areas relating to the use of index theory and spectral flow.