# Spectral Theory of Hyperbolic Surfaces Spring 2021

- Lecturer
- Claire Burrin
- Lectures
- Tuesdays, 2-4pm, on Zoom, starting Feb 23
- Contact me if you have not received the Zoom link!

### Content

The Laplacian plays a prominent role in many parts of mathematics. On a flat surface like the torus, understanding its spectrum is the topic of Fourier analysis, whose 19th century development allowed to solve the heat and wave equations. On the sphere, one studies spherical harmonics. In this course, we will study the spectrum of hyperbolic surfaces and its Maass forms (eigenfunctions). We will start from scratch, with an overview of hyperbolic geometry and harmonic analysis on the hyperbolic plane. The objectives are to prove the spectral theorem and Selberg's trace formula, and explore applications in geometry and number theory.

Tentative syllabus:

Hyperbolic geometry (the hyperbolic plane and Fuchsian groups)
Construction of arithmetic hyperbolic surfaces
Harmonic analysis on the hyperbolic plane
The spectral theorem
Selberg's trace formula
Applications in geometry (isoperimetric inequalities, geodesic length spectrum)
and number theory (links to the Riemann zeta function and Riemann hypothesis)
Possible further topics (if time permits):

Eisenstein series
Explicit constructions of Maass forms (after Maass)
A special case of the Jacquet-Langlands correspondence (after the exposition of Bergeron, see references)
Knowledge of the material covered in the first two years of bachelor studies is assumed. Prior knowledge of differential geometry, functional analysis, or Riemann surfaces is not required.

## Lectures

Links to the lectures contents and materials will be posted here.

Here are the typed lecture notes.

## References

We will most closely follow Bergeron's book (with deviations and add-ons!).

Nicolas Bergeron, The Spectrum of Hyperbolic Surfaces, Springer Universitext 2011.
Armand Borel, Automorphic forms on SL(2,R), Cambridge University Press 1997.
Peter Buser, Geometry and spectra of compact Riemann surfaces, Birkhäuser 1992.
Henryk Iwaniec, Spectral methods of automorphic forms. Graduate studies in mathematics, AMS 2002.
Atle Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20 (1956), p.47-87.