Spectral Theory of Hyperbolic Surfaces Spring 2021

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


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.


    Links to the lectures contents and materials will be posted here. Here are the typed lecture notes.
    Class Topics Notes Recording
    Feb 23 Notions of non-Euclidean and Riemannian geometry Class notes Recording
    Mar 2 Spectral theory of the Laplacian on the torus; geometric and arithmetic applications Class notes Recording
    Mar 9 Poisson summation and applications Class notes Recording
    Mar 16 Geometry of the hyperbolic plane Class notes Recording
    Mar 23 Metric properties of the hyperbolic plane; Fuchsian groups and hyperbolic surfaces Class notes Recording
    Mar 30 Fundamental domains and examples of Fuchsian groups Class notes Recording
    Apr 13 Geometric and algebraic features of fundamental domains. Spectral problem. Class notes Recording
    Apr 20 Point-pair invariants. Hilbert-Schmidt operators. Spectral theorem. Class notes Recording
    Apr 27 Spherical functions. Selberg transform. Solution to the heat equation. Class notes Recording
    May 4 Euclidean and hyperbolic circle problems. Spectrum of Laplacian. Class notes Recording
    May 11 Selberg's trace formula Class notes Recording
    May 18 Huber's theorem, Weyl's law Class notes Recording
    May 25 On the first nontrivial eigenvalue Class notes Recording
    Jun 1 Selberg's eigenvalue conjecture and expanders Class notes Recording


    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.