Claire Burrin: Katalogdaten im Frühjahrssemester 2021

NameFrau Dr. Claire Burrin
URLhttps://people.math.ethz.ch/~clburrin/
DepartementMathematik
BeziehungDozentin

NummerTitelECTSUmfangDozierende
401-4118-21LSpectral Theory of Hyperbolic Surfaces Information 4 KP2VC. Burrin
KurzbeschreibungThe 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).
LernzielWe 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.
InhaltTentative 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)
LiteraturNicolas 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.
Voraussetzungen / BesonderesKnowledge 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.