MATH 480/595/693/740: Hyperbolic Geometry and Automorphic Forms. Winter 2013


This is an undergraduate/graduate course, and mini-seminar, on hyperbolic geometry and automorphic forms. Students are expected to give many lectures in the course.
  • D. Jakobson
    Office: McGill, Burnside Hall, 1220
    Office Hours: TBA
    Tel: 398-3828
  • J. Toth
    Office: McGill, Burnside Hall, 1221
    Office Hours: TBA
    Tel: 398-3847

  • Organizational meeting: Thursday, January 10, 12 noon, Burnside Hall, Room 920 (9th floor)
  • Regular lectures will be held on Mondays, 14:30-16:30, Burnside 1120.
  • Lectures start Monday, January 14.
  • From January 21 onward, the lectures will be moved to Burnside 719A, 14:30-16:00.

    The two basic references for the course will be Svetlana Katok's book Fuchsian groups, and Peter Sarnak's book Some applications of modular forms. We shall discuss selected topics from both books. Additional references will be added.
    A course Dynamics and Quantum Chaos on hyperbolic surfaces by Alex Gorodnik at Bristol.
    Presentations, Grading
    The students registered for the course will be expected to make two oral presentations (between 30-50 minutes) on one of the topics suggested by the instructors. Graduate students will also be expected to write a short paper on a topic related to the course material. The grades will be based on the presentations and the paper.

    Homework problems: your grade will not depend on them, but we suggest that you should do a few problems to better understand the material:
  • S. Katok, Chapter 1: Problems 1.1, 1.4, 1.6, 1.10, 1.11, 1.12.

  • Topics for presentation (references to be added):
  • Selberg trace formula (several lectures);
  • Linking of geodesics on modular surface and other Riemann surfaces;
  • Ergodicity of geodesic flow;
  • Counting closed geodesics on Riemann surfaces;
  • Selberg 3/16 theorem (related also to the material in the reading course on expanders);
  • Multiplicity of length spectra for arithmetic surfaces;
  • Quantum ergodicity of Eisenstein series (likely D.J. will lecture on that);
  • Applications to Ruziewicz problem;
  • Geodesic flows. continued fractions and Diophantine approximations
  • Discrete subgroups of PSL(2,C) and the Mandelbrot set: a paper by R. Brooks and J.P. Matelski.
  • Small eigenvalues on Riemann surfaces
  • Teichmuller spaces
  • Hecke operators

  • Closed geodesics, continued fractions, etc: papers by S. Katok:
  • S. Katok, Coding of closed geodesics after Gauss and Morse, Geom. Dedicata 63 (1996), 123-145. pdf
  • S. Katok and I. Ugarcovici: Symbolic dynamics on the modular surface and beyond: Bull. of the Amer. Math. Soc., 44, no. 1 (2007), 87-132. pdf
  • S. Katok. Fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics. Lecture notes, 2007 Clay summer school in Pisa. pdf
  • S. Katok and I. Ugarcovici: Arithmetic coding of geodesics on the modular surface via continued fractions.
  • S. Katok: Elliptic operators and solutions of cohomological equations for geodesic flows with hyperbolic behavior.
  • Automorphic forms, Quantum chaos etc: papers by P. Sarnak:
  • P. Cohen and P. Sarnak. Notes on the Trace Formula: chapter 6 and chapter 7.
  • P. Sarnak. Notes on the Generalized Ramanujan Conjectures: pdf.
  • A. Gamburd, D. Jakobson and P. Sarnak. Spectra of elements in the group ring of SU(2), Jour. of Eur. Math. Soc. 1(1) (1999), 51-85. ps
  • P. Sarnak. Rademacher lectures. Equidistribution and primes. pdf
  • P. Sarnak, 2011 lectures. Mobius randomness and dynamics. pdf
  • P. Sarnak, 2010 lectures. Mass equidistribution and zeros/nodal domains of modular forms. pdf
  • Other papers:
  • D. Jakobson. Quantum Unique Ergodicity for Eisenstein Series on PSL(2,Z)\PSL(2,R). Annales de l'Institut Fourier, 44(5) (1994), 1477-1504. ps
  • D. Jakobson. Equidistribution of cusp forms on PSL(2,Z)\PSL(2,R). Annales de l'Institut Fourier, 47(3) (1997), 967-984. ps

  • 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