Minerva Distinguished Visitor Lectures at Princeton University

Let $X$ be a complete hyperbolic metric on a surface of genus $g$ with $n$ punctures. In this lecture I will discuss the problem of the growth of $s^{k}_{X}(L)$, the number of closed curves of length at most $L$ on $X$ with at most $k$ self-intersections. More generally, we investigate the properties of the orbit of an arbitrary closed curve $\gamma$ under the action of the mapping class group. I will also discuss problems regarding the distribution of the corresponding geodesics on $T^1(X)$.