In this talk we show that directions on flat surfaces which are poorly approximated by saddle connection directions are a winning set for Schmidt's game. This extends a result of Schmidt for the torus and strengthens a result of Kleinbock and Weiss. We will state some consequences of the result. This is joint work with Yitwah Cheung and Howard Masur.

For mapping class groups there is a notion of convex cocompactness, due to Farb and Mosher, defined by way of analogy with the concept of the same name in Kleinian groups. On the other hand, there are certain Kleinian groups which can themselves naturally be thought of as subgroups of mapping class groups. After describing some of the background, I will discuss a direct relationship between convex cocompactness in the two settings for this special class of groups. This is joint work with Spencer Dowdall and Richard Kent.

Thirty years ago Dennis Johnson proved that the Torelli group \(T_g\) in genus g is finitely generated for all \(g \ge 3\). Recently Putman initiated a program to give a new proof of Johnson's fundamental result by proving that if \(T_3\) is finitely generated, then so is \(T_g\) for all \(g > 3\). In this talk I will explain why \(T_3\) is finitely generated, which I will do in the more general context of understanding fundamental groups of a certain class of branched coverings of quasi-projective varieties having a complete Kaehler metric with non-positive sectional curvatures. I will also discuss the problem of determining whether \(T_3\) is finitely presented. The principal tool in the proof of the finite generation result and the investigation of \(H_2(T_3)\) is the stratified Morse theory of Goresky and MacPherson.

We will discuss recent joint work with Dave Futer in which we study hyperbolic 3-orbifolds having singular locus a link. We have identified the smallest volume hyperbolic 3-orbifold having base space the 3-sphere and singular locus a knot. We also identify the smallest volume hyperbolic 3-orbifold with base space any homology 3-sphere and singular locus a link. With weaker homology assumptions, we obtain a lower bound on the volume of a link orbifold.

Let S be a surface of genus g with n punctures. Let Mod(S) the mapping class group. It acts on T(S), the Teichmuller space of S by isometries with respect to the Teichmuller metric. The action is properly discontinuous. There are various counting problems associated with this action. One of which is the lattice counting problem which asks, given a pair of points x,y and a large number r, how many elements of Mod(S) send y into a ball of radius r centered at x. Athreyev, Bufetov, Eskin and Mirzakhani found precise asymptotics of the form \( ce^{(6g-6+n)r}\) for the number as r goes to infinity. Maher showed that "most" such elements are pseudo-Anosov. I will give a quantitative version by showing that the number that are not pseudo-Anosov have strictly smaller exponential growth.