Georgia Institute of Technology College of Sciences

We introduce a new surgery operation for contact manifolds called the Liouville connect sum. This operation -- which includes Weinstein handle attachment as a special case -- is designed to study the relationship between contact topology and symplectomorphism groups established by work of Giroux and Thurston-Winkelnkemper. The Liouville connect sum is used to generalize results of Baker-Etnyre-Van Horn-Morris and Baldwin on the existence of "monodromy multiplication cobordisms" as well as results of Seidel regarding squares of symplectic Dehn twists.

A permutation of the set {1,2,...,n} is connected if there is no k < n such that the set of the first k numbers is invariant as a set under the permutation. For each permutation, there is a corresponding graph whose vertices are the letters of the permutation and whose edges correspond to the inversions in the permutation. In this way, connected permutations correspond to connected permutation graphs. We find a growth process of a random permutation in which we start with the identity permutation on a fixed set of letters and increase the number of inversions one at a time. After the m-th step of the process, we obtain a random permutation s(n,m) that is uniformly distributed over all permutations of {1,2,...,n} with m inversions. We will discuss the evolution process, the connectedness threshold for the number of inversions of s(n,m), and the sizes of the components when m is near the threshold value. This study fits into the wider framework of random graphs since it is analogous to studying phase transitions in random graphs. It is a joint work with my adviser Boris Pittel.

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. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.

We will study one and two weight inequalities for several different operators from harmonic analysis, with an emphasis on vector-valued operators. A large portion of current research in the area of one weight inequalities is devoted to estimating a given operators' norm in terms of a weight's A_p characteristic; we consider some related problems and the extension of several results to the vector-valued setting. In the two weight setting we consider some of the difficulties of characterizing a two weight inequality through Sawyer-type testing conditions.