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.

Much of what is known about automorphisms of free groups is given by analogy to results on mapping class groups. One desirable result is the celebrated Nielson-Thurston classification of the mapping class group into reducible, periodic, or pseudo Anosov homeomorphisms. We will discuss attempts at analogous results for free group automorphisms.

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.