Georgia Institute of Technology College of Sciences

The hyperelliptic Torelli group SI(S) is the subgroup of the mapping class group of a surface S consisting of elements which commute with a fixed hyperelliptic involution and which act trivially on homology. The group SI(S) appears in a variety of settings, for example in the context of the period mapping on the Torelli space of a Riemann surface and also as a kernel of the classical Burau representation of the braid group. We will show that the cohomological dimension of SI(S) is g-1; this result fits nicely into a pattern with other subgroups of

There are two simple ways to construct new surface bundles over surfaces from old ones, namely, we can connect sum along the base or the fiber. In joint work with Inanc Baykur, we construct explicit surface bundles over surfaces that are indecomposable in both senses. This is achieved by first translating the problem into one about embeddings of surface groups into mapping class groups.

There is little known about the existence of contact strucutres in high dimensions, but recently in work of Casals, Pancholi and Presas the 5 dimensional case is largely understood. In this talk I will discuss the existence of contact structures on 5-manifold and outline an alternate construction that will hopefully prove that any almost contact structure on a 5-manifold is homotopic, though almost contact structures, to a contact structure.

Surface bundles and Lefschetz fibrations over surfaces constitute a rich source of examples of smooth, symplectic, and complex manifolds. Their sections and multisections carry interesting information on the smooth structure of the underlying four-manifold. In this talk we will discuss several problems and results on surface bundles, Lefschetz fibrations, and their (multi)sections, which we will tackle, for the most part, using various mapping class groups of surfaces.

We prove the moduli space M_{g,n} of the surface of g genus with n punctures admits no complete, visible, nonpositively curved Riemannian metric, which will give a connection between conjectures from P.Eberlein and Brock-Farb. Motivated from this connection, we will prove that the translation length of a parabolic isometry of a proper visible CAT(0) space is zero. As an application of this zero property, we will give a detailed answer toP.Eberlein's conjecture.

Let S_g be a closed orientable surface of genus g > 1 and C a simple closed nonseparating curve in S_g. Let t_C denote a left handed Dehn twist about C. A fractional power of t_C of exponent L/n is a h in Mod(S_g) such that h^n = t_C^L. Unlike a root of a t_C, a fractional power h can exchange the sides of C. We will derive necessary and sufficient conditions for the existence of both side-exchanging and side-preserving fractional powers. We will give some applications of the main result in both cases.

Recently, Sarkar showed that a smooth marked cobordism between two knots K_1 , K_2 induces a map between the knot Floer homology groups of the two knots HFK(K_1 ), HFK(K_2 ). It has been conjectured that this map is well defined (with respect to smooth marked cobordisms). After outlining what needs to be shown to prove this conjecture, I will present my current progress towards showing this result for the combinatorial version of HFK.

Berge has a construction that produces knots in S^3 that admit a lens space surgery. Conjecturally, his construction produces all such knots. This talk will consider knots that have such a surgery, and some of their contact geometric properties. In particular, knots in S^3 with a lens space surgery are fibered, and they all support the tight contact structure on S^3. From recent work of Hedden and Plamenevskaya, we also know that the dual to a lens space surgery on such a knot supports a tight contact structure on the resulting lens space.