Mathematics Colloquium

Let Y be a compact Riemann surface with a Riemannian metric of constant curvature -1. Consider the corresponding Laplace-Beltrami operator in the space of functions on Y and fix its eigenfunction $\phi$. Such function is called Maass form; study of such forms plays an important role in Geometry and Number Theory. I will introduce a generalmethod how to bound invariants arising from Maass forms. This method is based on representation theory of the group SL(2, R). I will discuss the following concrete problem. Fix a closed geodesic $C\subset Y$, consider the restriction f of the Maass form $$\phi to C and decompose it into its Fourier series $f=\sum a_n \exp(2\pi int)$. The problem is to give good bounds for Fourier coefficients an when n tends to infinity.