Georgia Institute of Technology College of Sciences

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds.

Legendrian contact homology is an invariant in contact geometry that assigns to each Legendrian submanifold a dg-algebra. While well-defined, it depends upon counts of holomorphic curves that can be hard to calculate in practice. In this talk, we introduce a class of Legendrian tori constructed as the product of collections of Legendrian knots. For this class, we discuss how to explicitly compute the dg-algebra invariant of the tori in terms of diagram projections of the constituent Legendrian knots.

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds.

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds.

A well known result of Giroux tells us that isotopy classes of contact structures on a closed three manifold are in one to one correspondence with stabilization classes of open book decompositions of the manifold. We will introduce a stabilization-invariant property of open books which corresponds to tightness of the corresponding contact structure. We will mention applications to the classification of contact 3-folds, and also to the question of whether tightness is preserved under Legendrian surgery.

We will briefly talk about the introduction to Thruston norm and fibered face theory. Then we will discuss polynomial invariants for fibered 3-manifolds, so called Teichmuller polynomials. I will give an example for a Teichmuller polynomial and by using it, determine the stretch factors (dilatations) of a family of pseudo-Anosov homeomorphisms.

We will explore the notion of surgery on transverse knots in contact 3-manifolds. We will see situations when this operation does or does not preserves properties of the original contact structure, and avenues for further research.

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds.

A conjecture of Ivanov asserts that finite index subgroups of the mapping class group of higher genus surfaces have trivial rational homology. Putman and Wieland use what they call higher Prym representations, which are extensions of the representation induced by the action of the mapping class group on homology, to better understand the conjecture. In particular, they prove that if Ivanov's conjecture is true for some genus g surface, it is true for all higher genus surfaces.

Given any surface, we can construct its curve complex by considering isotopy classes of curves on the surface. If the surface has boundary, we can construct its arc complex similarly, with isotopy clasess of arcs, with endpoints on the boundary. In 1999, Masur and Minsky proved that these complexes are hyperbolic, but the proof is long and involved. This talk will discuss a short proof of the hyperbolicity of the curve and arc complex recently given by Hensel, Przytycki, and Webb.