**Global Analysis**is the analysis of differential operators on manifolds with emphasis on invariant aspects. The most prominent example of a globally defined invariant of an elliptic differential operator on a compact manifold is its index. On a compact manifold it can be computed by a local formula, the famous Atiyah-Singer index formula. On noncompact manifolds there are other interesting invariants. Global Analysis on non-compact manifolds also deals with inverse scattering problems and inverse spectral problems. These become increasingly important in physics and medicine.-
**QFT on curved spacetimes**is the so called semiclassical approximation to Quantum gravity. Free Quantum fields can be defined on any globally hyperbolic spacetime using the theory of general hyperbolic equations. Physical states are believed to satisfy the Hadamard property, that is their n-point distributions should satisfy a condition that restricts their wavefront set. Using these concepts the theory analysis perturbative and non-perturbative aspects of QFT interacting with a classical gravitational field. -
**Spectral Geometry and Quantum Chaos**investigates the spectrum of geometric operators like Dirac or Laplace operators. The relation between eigenfunctions and geometry is inspired by Quantum physics and has become a fruitful subject in pure mathematics. In the high energy or the semiclassical regime one expects to recover some of the classical features of the theory. That is there is a relation between the spectral properties and classical dynamical systems like the geodesic flow. Quantum Chaos deals with the question as to what extent one can still see choatic features of a classical system on the spectral side.