Georgia Institute of Technology College of Sciences

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact r surgery is symplectically/Stein fillable when r is in (0,1].

Understanding contact structures on hyperbolic 3-manifolds is one of the major open problems in the area of contact topology. As a first step, we try to classify tight contact structures on a specific hyperbolic 3-manifold. In this talk, we will review the previous classification results and classify tight contact structures on the Weeks manifold, which has the smallest hyperbolic volume. Finally, we will discuss how to generalize this method to classify tight contact structures on some other hyperbolic 3-manifolds.

The theorem of Dehn-Nielsen-Baer says the extended mapping class group is isomorphic to the outer automorphism group of the fundamental group of a surface. This theorem is a beautiful example of the interconnection between purely topological and purely algebraic concepts. This talk will discuss the background of the theorem and give a sketch of the proof.

We will discuss a celebrated theorem of Sharkovsky: whenever a continuous self-map of the interval contains a point of period 3, it also contains a point of period n , for every natural number n.

Assuming some "compatibility" conditions between a Riemannian metric and a contact structure on a 3-manifold, it is natural to ask whether we can use methods in global geometry to get results in contact topology. There is a notion of compatibility in this context which relates convexity concepts in those geometries and is well studied concerning geometry questions, but is not exploited for topological questions. I will talk about "contact sphere theorem" due to Etnyre-Massot-Komendarczyk,

Three dimensional lens spaces L(p,q) are well known as the first examples of 3-manifolds that were not known by their homology or fundamental group alone. The complete classification of L(p,q), upto homeomorphism, was an important result, the first proof of which was given by Reidemeister in the 1930s. In the 1980s, a more topological proof was given by Bonahon and Hodgson. This talk will discuss two equivalent definitions of Lens spaces, some of their well known properties, and then sketch the idea of Bonahon and Hodgson's proof.

I will introduce the notion of satellite knots and show that a knot in a 3-sphere is either a torus knot, a satellite knot or a hyperbolic knot.

There are a number of ways to define the braid group. The traditional definition involves equivalence classes of braids, but it can also be defined in terms of mapping class groups, in terms of configuration spaces, or purely algebraically with an explicit presentation. My goal is to give an informal overview of this group and some of its subgroups, comparing and contrasting the various incarnations along the way.

We will discuss the relationship between diffeomorphis groups of spheres and balls. And try to give an idea of existense of exotic structures on spheres.

Lagrangian fillings of Legendrian knots are interesting objects that are related, on one hand, to the 4-genus of the underlying smooth knot and, on the other hand, to Floer-type invariants of Legendrian knots. Most work on Lagrangian fillings to date has concentrated on orientable fillings. I will present some first steps in constructions of and obstructions to the existence of (decomposable exact) non-orientable Lagrangian fillings.