MATH 595: Spectral theory of automorphic forms. Winter 2014.

D. Jakobson
Office: BH1220
Office Hours: Tuesday and Wednesday, 13:30-14:30
Tel: 398-3828
Web Page:
Course page:
  • Organizational meeting: Wednesday, January 8, Burnside 1120, 12noon
  • Lectures: starting January 14
  • Tuesday, 11:30-13:00, Burnside 1234
  • Wednesday, 11:30-13:00, Burnside 1120
  • There will be no lecture on Wednesday, January 15. There will be a lecture on Friday, January 17, from 10:00-11:30, in Burnside 1205.
  • Presentations will be held on April 8 and 9 at usual times
  • Extra presentations will be held on Friday, April 11; Lounge on the 10th floor of Burnside is reserved from 10:30-12:30.
  • The last day for presentations will be Monday, April 14. Burnside 1205 has been reserved from 10am-12noon.

    In the course we shall discuss spectral theory of automorphic forms for discrete co-compact or finite-area subgroups of SL(2,R). We shall follow the book "Spectral theory of automorphic forms" by H. Iwaniec. We expect to consider some of the following topics (as time permits): harmonic analysis in the hyperbolic plane, Fuchsian groups, Eisenstein series, cusp forms, the spectral theorem, Rankin-Selberg method, Selberg trace formula, counting closed godesics, Weyl's law, applications to number theory.
  • Required: H. Iwaniec. Spectral methods of automorphic forms. Second edition. Graduate Studies in Mathematics, 53. American Mathematical Society, Providence, RI; Revista Matematica Iberoamericana, Madrid, 2002. ISBN: 0-8218-3160-7
  • Additional references: Svetlana Katok, Fuchsian groups; Peter Sarnak, Some applications of modular forms.

  • Presentations, Grading
    In addition to doing homework assignments, the students registered for the course will be expected to give one or two oral presentations (around 30 minutes) on a topic suggested by the instructor. The grade will be based on the assignments and on these presentations.

    Previous related courses at McGill
  • Winter 2013: Seminar on Hyperbolic Geometry and Automorphic Forms. This web page has links to some results in geometry and dynamical systems that are only sketched in H. Iwaniec's book.

  • Possible themes for Presentation
  • Estimates for Fourier coefficients of Mass forms (for 1 or 2 people; Iwaniec, Chapter 8)
  • Spectral Theory of Kloosterman sums (1 or 2 people; Iwaniec, chapter 9)
  • Selberg Trace formula (2 people; Iwaniec, chapter 10)
  • Weyl's law for hyperbolic manifolds; lattice point counting (2 people; Iwaniec, Chapter 11-12)
  • Kleinian groups (1 or 2 people), topics to be discussed with instructor
  • Work of Brooks and Matelski: Mandelbrot set and Kleinian groups
  • Applications to spectral gaps for subgroups of the rotation group; distribution of lattice points on spheres (work of Lubotzky-Phillips-Sarnak; 1 or 2 people; explained in Sarnak's book)

  • Links

  • Course Dynamics and Quantum Chaos on hyperbolic surfaces by Alex Gorodnik at Bristol.
  • A book by Peter Buser (in Google books): Geometry and Spectra of Compact Riemann Surfaces
  • Shlomo Sternberg: Lecture notes in Real Analysis

  • NOTICE: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see McGill web page on Academic Integrity for more information).
    NOTICE: In accord with McGill University's Charter of Student Rights, students in this course have the right to submit in English or in French any work that is to be graded.
    NOTICE: In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change.