Georgia Institute of Technology College of Sciences

Following up on the previous series of talks we will show how to construct Lagrangian Floer homology and discuss it properties.

This is joint work with Mike Sullivan. We consider a Legendrian surface L in R5 or more generally in the 1-jet space of a surface. Such a Legendrian can be conveniently presented via its front projection which is a surface in R3 that is immersed except for certain standard singularities. We associate a differential graded algebra (DGA) to L by starting with a cellular decomposition of the base projection to R2 of L that contains the projection of the singular set of L in its 1-skeleton.

There is no general h-principle for Legendrian embeddings in contact manifolds. In dimension 3, however, Legendrian knots in the complement of an overtwisted disc, which are called loose, satisfy an h-principle. We will discuss the high dimensional analog of loose knots.

There is no general h-principle for Legendrian embeddings in contact manifolds. In dimension 3, however, Legendrian knots in the complement of an overtwisted disc, which are called loose, satisfy an h-principle. We will discuss the high dimensional analog of loose knots.

We will discuss a way of explicitly constructing ribbon knots using one-two handle canceling pairs. We will also mention how this is related to some recent work of Yasui, namely that there are infinitely many knots in (S^3, std) with negative maximal Thurston-Bennequin invariant for which Legendrian surgery yields a reducible manifold.

In this talk, I will define Conley-Zehnder index of a periodic Reeb orbit and will give several characterizations of this invariant. Conley-Zehnder index plays an important role in computing the dimension of certain families of J-holomorphic curves in the symplectization of a contact manifold.

Normal rulings are decompositions of a projection of a Legendrian knot or link. Not every link has a normal ruling, so existence of a normal ruling gives a Legendrian link invariant. However, one can use the normal rulings of a link to define the ruling polynomial of a link, which is a more useful Legendrian knot invariant. In this talk, we will discuss normal rulings of Legendrian links in various manifolds and prove that the ruling polynomial is a Legendrian link invariant.

A 2-knot is defined to be an embedding of S^2 in S^4. Unlike the theory of concordance for knots in S^3, the theory of concordance of 2-knots is trivial. This talk will be framed around the related concept of 0-concordance of 2-knots. It has been conjectured that this is also a trivial theory, that every 2-knot is 0-concordant to every other 2-knot. We will show that this conjecture is false, and in fact there are infinitely many 0-concordance classes. We'll in particular point out how the concept of 0-concordance is related to understanding smooth structures on S^4.