### 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.