Georgia Institute of Technology College of Sciences

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.

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).

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

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

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.

The outer automorphism group Out(F) of a non-abelian free group F of finite rank shares many properties with linear groups and the mapping class group Mod(S) of a surface, although the techniques for studying Out(F) are often quite different from the latter two. Motivated by analogy, I will present some results about Out(F) previously well-known for the mapping class group, and highlight some of the features in the proofs which distinguish it from Mod(S).

We will prove a duality between locally compact Hausdorff spaces and the C*-algebra of continuous complex-valued functions on that space. Formally, this is the equivalence of the opposite category of commutative C*-algebras and the category of locally compact Hausdorff spaces.

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.