Random Methods in 3-manifold theory

In this talk we will present an application of random methods (by considering random walks in mapping class groups) to showing existence of certain 3-manifolds without giving any explicit construction. More specifically, we show that for any given integers $g, n$ with $g \geq 2$ there exist closed hyperbolic 3-manifolds with Heegaard genus $g$ and Casson invariant $n$. This is joint work with Alex Lubotzky.