Organizers: Filip Pietrzak, Ivan Yakovlev

Time: Every Tuesday 15.00-16.00 (tbc)

Place: MPIM Seminar Room

The aim of this reading group is to understand the connections between the spectrum of the Laplacian on a Riemannian manifold and its geometry. The reading group will be divided into two parts. The first one (~ 6 talks) will focus on classical results (heat kernel, eigenvalue inequalities, length spectrum, Selberg trace formula and isospectrality) with hyperbolic surfaces as a primary example. The second part will consist of talks about modern research topics in the area, chosen by the participants. These might include: Selberg conjecture, spectra of random surfaces, expander graphs, Sarnak conjecture, quantum ergodicity, Laplacian acting on differentials, heat kernel approach to Atiyah-Singer theorem...

References

• Buser - Geometry and Spectra of Compact Riemann Surfaces
• Chavel - Eigenvalues in Riemannian Geometry
• Rosenberg - The Laplacian on a Riemannian manifold
• Levitin, Mangoubi, Polterovich - Topics in Spectral Geometry
• Bergeron - The Spectrum of Hyperbolic Surfaces
• Berger, Gauduchon, Mazet - Le Spectre d'une Variété Riemannienne

Schedule

• 28/04, Ivan Yakovlev
  Basic examples, spectral theorem via heat kernel, existence of heat kernel on compact manifolds, Manikshisundaram-Pleijel expansion, some applications.