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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.

Series: Geometry Topology Seminar

The curve complex C(S) of a closed orientable surface S of genusg is an infinite graph with vertices isotopy classes of essential simpleclosed curves on S with two vertices adjacent by an edge if the curves canbe isotoped to be disjoint. By a celebrated theorem of Masur-Minsky, thecurve complex is Gromov hyperbolic. Moreover, a pseudo-Anosov map f of Sacts on C(S) as a hyperbolic isometry with "north-south" dynamics and aninvariant quasi-axis. One can define an asymptotic translation length for fon C(S). In joint work with Chia-yen Tsai, we prove bounds on the minimalpseudo-Anosov asymptotic translation lengths on C(S) . We shall alsooutline related interesting results and questions.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

I will talk about the asymptotic geometry of Teichmuller space equipped with the Weil-Petersson metric. In particular, I will give a criterion for determining when two points in the asymptotic cone of Teichmuller space can be separated by a point; motivated by a similar characterization in mapping class groups by Behrstock-Kleiner-Minsky-Mosher and in right angled Artin groups by Behrstock-Charney. As a corollary, I will explain a new way to uniquely characterize the Teichmuller space of the genus two once punctured surface amongst all Teichmuller space in that it has a divergence function which is superquadratic yet subexponential.

Series: Geometry Topology Seminar

I will discuss the following geometric problem. If you are
given an abstract 2-dimensional simplicial complex that is homeomorphic to a
disk, and you want to (piecewise linearly) embed the complex in the plane so
that the boundary is a geometric square, then what are the possibilities
for the areas of the triangles?
It turns out that for any such simplicial complex there is a
polynomial relation that must be satisfied by the areas. I will report on joint work with
Jamie Pommersheim in which we attempt to understand various features of this
polynomial, such as the degree. One thing we do not know, for
instance, if this degree is expressible in terms of other known integer invariants
of the simplicial complex (or of the underlying planar graph).

Series: Geometry Topology Seminar

We will discuss SL(N,C) representations of 3-manifolds, and their complex volumes, theoretically and computationally.

Series: Geometry Topology Seminar

The Neumann-Zagier equations are well-understood objects of classical hyperbolic geometry. Our discovery is that they have a nontrivial quantum content, (that for instance captures the perturbation theory of the Kashaev invariant to all orders) expressed via universal combinatorial formulas. Joint work with Tudor Dimofte.