Jeff Brock - TBA

Richard Kent - Thoughts on convex cocompactness in mapping class groups

I will talk about some ideas related to Farb and Mosher's notion of convex cocompactness in mapping class groups.

Robert Tang - Shadows of Teichmueller discs in the curve graph (joint with Richard Webb)

Given a flat structure on a surface, we can deform the metric using elements of SL(2,R) to obtain different metrics. The SL(2,R)-orbit under this action is called a Teichmueller disc. The set of systoles arising on some flat metric on a Teichmueller disc gives a quasiconvex subset in the curve graph associated with the surface. We will discuss ways of finding nearest point projections to the systole set in the curve complex. We will also describe other sets of curves associated to a given Teichmueller disc.

Qiongling Li - Asymptotic Behavior of Certain Families in Hitchin Components

Hitchin components for $SL(n,R)$ is the component in the space of representations into $SL(n,R)$ which can deform to Fuchsian locus. In this talk, I will first go through basics of Higgs bundles and construction of Hitchin components for $SL(n,R)$ inside the moduli space of Higgs bundles. I will then introduce recent work with Brian Collier on asymptotic behaviors of certain families in Hitchin components. Namely, consider the family of Higgs bundles $(\mathcal{E},t{\phi})$, we try to analyze the asymptotic behavior of the corresponding representation $\rho_t$ as $t\rightarrow \infty$ in two special cases.

Kathryn Lindsey - Translation surfaces and the horocycle flow

A translation surface is, roughly speaking, a surface built from a finite collection of polygons made of graph paper by gluing the edges of the polygons together according to a specific set of rules. An element of $SL(2,\mathbb{R})$ acts on a translation surface by affinely stretching each of the polygons that make up the surface - resulting in a new translation surface. The horocycle flow is the action of the subgroup of An element of $SL(2,\mathbb{R})$ consisting of matrices of the form $\bigl(\begin{smallmatrix} 1&t\\ 0&1 \end{smallmatrix} \bigr)$, $t \in \mathbb{R}$. How is the orbit of a translation surface under the horocycle flow related to the orbit of that surface under all of $SL(2,\mathbb{R})$? It turns out that for any translation surface, after first rotating the surface by any one of a large set of angles, the horocycle orbit closure equals the $SL(2,\mathbb{R})$ orbit closure! This in turn leads to a characterization of lattice surfaces (translation surfaces whose orbits are closed) in terms of the horocycle flow. Based on joint work with Jon Chaika.

Piotr Przytycki - Arcs intersecting at most once

We prove that on a punctured oriented surface with Euler characteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a closed surface.

Hongbin Sun - Virtual domination of 3-manifolds

For any closed oriented hyperbolic 3-manifold M, and any closed oriented 3-manifold N, we will show that M admits a finite cover M', such that there exists a degree-2 map f : M'-> N, i.e. M virtually 2-dominates N.

Bena Tshishiku - Point-pushing and Nielsen realization

Let M be a manifold with mapping class group Mod(M). Any subgroup G < Mod(M) can be represented by a collection of diffeomorphisms that form a group up to isotopy. The Nielsen realization problem asks whether or not G can be represented as an honest subgroup of diffeomorphisms. We will discuss a special case of this problem when M is a locally symmetric manifold and G \simeq pi_1(M) is the "point-pushing" subgroup. This generalizes work of Bestvina-Church-Souto.