Georgia Institute of Technology College of Sciences

We prove that the torsion of any smooth closed curve in Euclidean space which bounds a simply connected locally convex surface vanishes at least 4 times (vanishing of torsion means that the first 3 derivatives of the curve are linearly dependent). This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in 3-space.

There is a beautiful idea that one can study spaces by studying associated geometric objects. More specifically one can associate to a manifold (that is some space) a symplectic or contact manifold (that is the geometric object). The question is how useful is this idea. We will discuss this idea and related questions for subspaces (that is immersions and embeddings) with a focus on curves in the plane and knots in three space. If time permits we will discuss powerful new tools from contact geometry that allow one use this idea to construct

This is the fourth of several talks discussing embeddings of manifolds. I will discuss some general results for smooth manifolds, but focus on embeddings of contact manifolds into other contact manifolds. Particular attention will be paid to embeddings of contact 3-manifolds in contact 5-manifolds. I will discuss two approaches to this last problem that are being developed jointly with Yanki Lekili.

This is the second of several talks discussing embeddings of manifolds. I will discuss some general results for smooth manifolds, but focus on embeddings of contact manifolds into other contact manifolds. Particular attention will be paid to embeddings of contact 3-manifolds in contact 5-manifolds. I will discuss two approaches to this last problem that are being developed jointly with Yanki Lekili.

In joint work with Vera Vertesi, we extend the functoriality in Heegaard Floer homology by defining a Heegaard Floer invariant for tangles which satisfies a nice gluing formula. We will discuss theconstruction of this combinatorial invariant for tangles in S^3, D^3, and I x S^2. The special case of S^3 gives back a stabilized version of knot Floer homology.

This is the first of several talks disussing embeddings of manifolds. I will discuss some general results for smooth manifolds, but focus on embeddings of contact manifolds into other contact manifolds. Particular attentaion will be payed to embeddings of contact 3-manifolds in contact 5-manifolds. I will discuss two approaches to this last problem that are being developed jointly with Yanki Lekili.

In this talk we consider the contact embeddings of contact 3-manifolds to S^5 with the standard contact structure.Every closed 3-manifold can be embedded to S^5 smoothly by Wall's theorem. The only known necessary condition to a contact embedding to the standard S^5 is the triviality of the Euler class of the contact structure.