An index theorem on asymptotically static spacetimes with compact Cauchy surface

Prof Michał Wrochna (Cergy Paris Université) will present his research in Analysis.

A theorem due to Bär and Strohmaier (Amer. J. Math., 141 (5)) says that the Dirac operator on a Lorentzian manifold with compact Cauchy surface is Fredholm if Atiyah–Patodi–Singer boundary conditions are imposed at finite times. Furthermore, the index is given by a geometric formula that parallels as closely as possible the Atiyah–Patodi–Singer theorem in the Riemannian setting. In this talk I will report on joint work with Dawei Shen (Sorbonne Université) which extends this result to the infinite-time setting. Furthermore, we demonstrate that in the infinite-time situation, Fredholm inverses are Feynman parametrices in the sense of Duistermaat–Hörmander, a property which allows to show relationships with local aspects of the geometry by a recent result of Bär–Strohmaier (arXiv:2012.01364).

The talks are held using Zoom. Anyone interested in attending (outside the School of Mathematics) should email Ben Sharp at b.g.sharp@leeds.ac.uk to request zoom coordinates