I am a postdoctoral researcher at the University of Tübingen, Department of Mathematics.
The field of sepctral geometry concerns with the connections between the geometry of manifolds and the spectrum of differential operators. The spectrum of the Laplace operator plays a crucial role in the inverse spectral problems. The most famous question relative to these problems was posed by Marc Kac in mid-60's :
" Can one hear the shape of a drum ? "
The answer is not always positive, in particular when we deal with manifolds with singularities. This question can be alternatively expressed as:
"How can one obtain information about the geometry of a manifold, such as the volume, the curvature, or the length of the closed geodesics, provided that we can
study the spectrum of certain differential operators? "
Harmonic analysis on locally symmetric spaces provides a poweful machinery in studying various invariants, such as the analytic torsion, as well as the dynamical zeta functions of Ruelle and Selberg.