Course Information and Syllabus

Topics in Analysis: SL(2,R)/Gamma

Please consult this course homepage for updates on the course schedule, texts, exercises, handouts...

Course title

Topics in Analysis: SL(2,R)/Gamma

Course number

Meeting times

110.726

It is scheduled for T Th 10: 30 - 12, but it will almost certainly change to M W 10: 30 - 12. Now in MD 309 Stay posted.

Professor:

Office Hours

Steve Zelditch

M  2  - 3PM  in Krieger 406 and by appointment.

Course description

SL(2, R) is the isometry group of the Poincare upper half plane H^2, and its quotient by a discrete subgroup Gamma is the unit tangent bundle of  the hyperbolic surface H^2/Gamma.  If one decomposes L^2(SL(2,R)/Gamma) into irreducible representations, one finds that the principal series correspond to eigenvalues of the Laplacian of the hyperbolic surface H^2/Gamma,  while the discrete series correspond to holomorphic line bundles over this Riemann surface. The representation theory of SL(2, R) has many applications to geometry and analysis on H^2/Gamma: for instance, it was used by Gelfand et al to prove that the geodesic flow is ergodic. The Selberg trace formula establishes a connection between lengths of closed geodesics and eigenvalues of the Laplacian.  Fourier coefficients of modular forms in the case Gamma = SL(2, Z) are important in number theory.  And the Laplacian is the basic model of Quantum Chaos. We will emphasize these topics.

Pre-requisites             It would help to have a little bit of  acquaintance with: Riemannian manifolds and geodesics, Lie groups, unitary representations and spectral theory We will go over all of the relevant material.

 1. HW 1

 2. HW 2

3.  HW3

4.  HW4

5.  HW5

Textbooks

For unitary representations of SL(2,R) we will use relevant sections of

1. A. W. Knapp,{\it Representation theory of semisimple groups. An overview based on examples},Princeton Landmarks in Mathematics. Princeton University Press, Princeton,
NJ, 2001. (This book is well written and has exercises).

2. R. Howe and E.C. Tan, {\it Nonabelian harmonic analysis. Applications of ${\rm SL}(2,R)$.} Universitext. Springer-Verlag, New York, 1992.

3. S. Lang, ${\rm SL}\sb 2(R)$, Graduate Texts in Mathematics, 105.
Springer-Verlag, New York, 1985. (This one is almost unreadable but has some useful material).

4.  M. E. Taylor, {\it Noncommutative harmonic analysis.}
Mathematical Surveys and Monographs, 22. American Mathematical Society, Providence, RI, 1986.

For quotients by discrete groups Gamma and the Selberg Trace formula, we will use sections of

1. P. J. Cohen and P. Sarnak, Notes.(they are online at Sarnak's homepage).

2. D. A. Hejhal, {\it The Selberg trace formula for ${\rm PSL}(2,R)$. Vol. I.}
Lecture Notes in Mathematics, Vol. 548. Springer-Verlag, Berlin-New York, 1976.

3. H. P. McKean, {\it Selberg's trace formula as applied to a compact Riemann surface}, Comm. Pure Appl. Math. {\bf 25} (1972), 225--246.
 

4.  A. B. Venkov, {\it Spectral theory of automorphic functions}, A
translation of Trudy Mat. Inst. Steklov. 153 (1981). Proc. Steklov
Inst. Math. 1982, no. 4(153), ix+163 pp. (1983).

Another reference is:

1. S. Helgason, {\it Topics in harmonic analysis on homogeneous spaces},
Progress in Mathematics, 13. Birkh{\"a}user, Boston, Mass., 1981.