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 AtiyahSinger index
formula. On noncompact manifolds there are other interesting
invariants. Global Analysis on noncompact manifolds also
deals with inverse scattering problems and inverse spectral
problems. These become increasingly important in physics and
medicine.


QFT on curved spacestimes 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 npoint distributions should satisfy a condition
that restricts their wavefront set. Using these concepts the
theory analysis perturbative and nonperturbative aspects of QFT
interacting with a classical gravitational field.
