20 april 2006, Andrew Booker (University of Michigan): Convergent Hejhal-type algorithms

Abstract. There have been many numerical investigations of the spectrum of the Laplace operator on non-compact, finite volume hyperbolic surfaces. The algorithm of D. Hejhal has proven to be robust and has yielded good results in many cases. However, there is no rigorous proof of either the convergence of the algorithm or that the results it gives are correct. In the talk I will give an overview of Hejhal's algorithms for the case of the modular group, and discuss some recent joint work with A. Strömbergsson and A. Venkatesh in which we compute and prove correct the first few eigenvalues to high precision. I will then show how to adapt the method to give algorithms similar to those of Hejhal, but for which one can prove convergence. If time permits I will discuss some related questions concerning large eigenvalue computations.