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Spectral Theory of Hyperbolic Surfaces
Last Updated: 2026-02-05 15:54:10
Abstract
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).
Objective
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.
Content
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)
Resources
Literature
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.
Learning Materials (Links)
- Main link
- Information
General Information
- Language
- English
- Levels
- BSC , MSC
Examination
- Type
- session examination
- Mode
- oral 20 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Spectral Theory of Hyperbolic Surfaces |
|
2 h weekly |
Offered In
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Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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