Scalar curvature is the simplest of several inequivalent notions of intrinsic curvature of higher dimensional Riemannian manifolds. The question what conditions on the scalar curvature tell us about the global shape of the manifold has been studied for many decades but it has gained enormous momentum by Gromov's Four lectures on scalar curvature which introduced novel concepts and techniques that have broadened the scope of inquiry in this field. We contribute to this very active research area, with an emphasis on Dirac operator methods and spin geometry.

For compact Riemannian manifolds with boundary we have a fairly complete understanding of boundary value problems for elliptic differential operators of first order. Dropping compactness leads to subtle questions which are currently investigated. The index theorem on Lorentzian manifolds mentioned above is concerned with a boundary value problem where the boundary is Riemannian. In this situation, it is not yet fully understood which boundary conditions are admissible. Moreover, if the boundary is timelike or a combination of timelike and spacelike many fundamental questions are still open.

The Atiyah-Singer index theorem and its refinement for manifolds with boundary due to Atiyah, Patodi, and Singer belong to the most significant mathematical achievements of the 20th century.  It computes the index of an elliptic differential operator on a manifold in geometric and topological terms and has numerous applications. We are interested in index theory on Lorentzian manifolds where hyperbolic operators need to be studied. While there is no Lorentzian analog of the Atiyah-Singer index theorem for closed manifolds, it turned out that there is one for the Atiyah-Patodi-Singer index theorem for manifolds with boundary. This has already found applications in quantum field theory. Currently, we are developing Lorentzian index theory further with an eye on applications in Lorentzian geometry.

The study of wave equations on curved spacetimes is a crucial area of research in mathematical physics, particularly in the context of general relativity and quantum field theory. Wave equations, which describe the propagation of fields such as electromagnetic waves, gravitational waves, and scalar fields, take on a more complex form when the underlying spacetime is curved. This curvature, which represents the influence of gravity according to Einstein's theory of general relativity, significantly impacts how these waves propagate and interact.

The study of wave equations on curved spacetimes presents significant mathematical challenges, leading to advances in differential geometry and partial differential equations. These equations often require sophisticated techniques to handle existence and uniqueness, singularities, asymptotic behaviors, and stability issues.