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Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

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
the mapping class group, particularly those of the Johnson filtration.
This is joint work with Leah Childers and Dan Margalit.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

I will talk briefly about how the study of fibred knots and Thurston's classification of automorphisms of surfaces in the 70's lead to Gabai and Oertel's work on essential laminations in the 80's. Some of this structure, for instance fractional Dehn twist coefficients, has implications in contact topology. I will describe results and examples, both old and new, that emphasize the special nature of S^3. This talk is based on joint work with Rachel Roberts.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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. Finally, we give a complete classification of a certain class of side-preserving and side-exchanging fractional powers on S_5.

Series: Geometry Topology Seminar

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. Conversely, we will use geometric arguments to obtain some structural results for mapping class groups.

Series: Geometry Topology Seminar

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. Specifically, I will present a generalization of Carter and Saito's movie move theorem to grid diagrams, give a very brief introduction to combinatorial knot Floer homology and then present a couple of the required chain homotopies needed for the proof of the conjecture.

Series: Geometry Topology Seminar

We find decomposition series of length at most two
for modular representations in characteristic p of mapping class groups of
surfaces induced by an integral version of the Witten-Reshetikhin-Turaev SO(3)-TQFT at the p-th root of unity.
This joint work with Gregor Masbaum.

Series: Geometry Topology Seminar

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. We consider the knots that are dual to Berge's knots, and we investigate whether the tight contact structure they support is a universally tight structure. Our results indicate a relationship between supporting this universally tight structure and being dual to a torus knot.