TCC Course — Fall 2012
The study of automorphic forms is a rich area of research. From its very nature it belongs to analytic number theory and addresses fundamental questions such as hyperbolic lattice point problems, and the generalized Riemann Hypothesis. The spectral theory of automorphic forms is also of importance in mathematical physics and includes a number of beautiful applications in ergodic theory and quantum chaos. Further applications address the large scale geometry and topology of our Universe.
The course will provide a gentle introduction into the spectral theory of automorphic forms. During the lectures, we will give the definitions of Fuchsian groups, modular and congruence subgroups, finite hyperbolic Riemann surfaces, invariant operators, automorphic integral kernels, Eisenstein series, Maass cusp forms, and the automorphic Green function. We will then prove the analytic continuation of Eisenstein series, Fourier expansion of Maass cusp forms, some spectral theorems, and will culminate in the Selberg trace formula which establishes a quantitative connection between the spectrum and the geometry of the Riemann surface Γ\H. We plan to cover the following:
Copies of the lectures will be posted here week by week after each course.
Lectures 1a, 1b, 2, 3, 4 & 5a, 5b, 6a, 6b, 7a, 7b, 8, 9.
The course will be assessed by solutions to the problem sheets. The problem sheets will be posted here and will be updated as the course goes on.