We prove a local version of the index theorem for Dirac-type operators on globally hyperbolic Lorentzian manifolds with Cauchy boundary. In case the Cauchy hypersurface is compact, we do not assume self-adjointness of the Dirac operator on the spacetime or of the associated elliptic Dirac operator on the boundary.In this case, integration of our local index theorem results in a generalization of previously known index theorems for globally hyperbolic spacetimes that allows for twisting bundles associated with non-compact gauge groups. This is joint work with Alexander Strohmaier.[-]

Liouville conformal field theory is a 2 dimensional field theory introduced in physics in the 80's. Here we give a probabilistic construction of the amplitudes of Riemann surfaces with boundary for this field theory, and we prove that they satisfy the so called Segal Axioms. This allows to decompose the correlation function of the theory using the diagonalisation of a certain operator, the Hamiltonian, using scattering theory. One can then show the formulas conjectured in physics in terms of conformal blocks. The spectral analysis, although in an infinite dimensional setting, has some similarities with scattering theory on symmetric spaces. This is joint work with Kupiainen, Rhodes and Vargas.[-]

I shall explain how to define and investigate primary and secondary invariants of G-invariant Dirac operators on a cocompact G-proper manifold, with G a connected real reductive Lie group. This involves cyclic cohomology and Ktheory. After treating the case of cyclic cocycles associated to elements in the differentiable cohomology of G I will move to delocalized cyclic cocycles, in particular, I will explain the challenges in defining the delocalized eta invariant associated to the orbital integral defined by a semisimple element g in G and in showing that such an invariant enters in an Atiyah-Patodi-Singer index theorem for cocompact G-proper manifolds. I will then consider a higher version of these results, based on the Song-Tang higher orbital integrals associated to a cuspidal parabolic subgroup P¡G with Langlands decomposition P=MAN and a semisimple element g in M. This talk is based on articles with Hessel Posthuma and with Hessel Postrhuma, Yanli Song and Xiang Tang.[-]

A complete Riemannian manifold is called asymptotically hyperbolic if its ends are modeled on neighborhoods of infinity in hyperbolic space. There is a notion of mass for this class of manifolds defined as a coordinate invariant computed in a fixed asymptotically hyperbolic end and measuring the leading order deviation of the geometry from the background hyperbolic metric in the end. Asymptotically hyperbolic manifolds arize naturally in mathematical general relativity, in particular, as slices of asymptotically Minkowski spacetimes, in which case the mass of the slice coincides with the Bondi mass of the spacetime. Having reviewed these and related concepts, we will discuss our proof of the positive mass theorem in the asymptotically hyperbolic setting, which relies on the original ideas of Schoen and Yau and involves a blow-up analysis of the so-called Jang equation, a geometric PDE of mean curvature type.[-]