Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

One of the general methods of proving h-principle is holonomic aprroximation. In this series of talks, I will give a proof of holonomic approximation theorem, and talk about some of its applications.

The Oka-Grauert principle is one of the first examples of an h-principle.  It states that for a Stein domain X and a complex Lie group G, the topological and holomorphic classifications of principal G-bundles over X agree.  In particular, a complex vector bundle over X has a holomorphic trivialization if and only if it has a continuous trivialization.  In these talks, we will discuss the complex geometry of Stein domains, including various characterizations of Stein domains, the classical Theorems A and B, and the Oka-Grauert principle.

Understanding contact structures on hyperbolic 3-manifolds is one of the major open problems in the area of contact topology. As a first step, we try to classify tight contact structures on a specific hyperbolic 3-manifold. In this talk, we will review the previous classification results and classify tight contact structures on the Weeks manifold, which has the smallest hyperbolic volume. Finally, we will discuss how to generalize this method to classify tight contact structures on some other hyperbolic 3-manifolds.

In this series of talks, we will study the relationship between the Alexander module and the bordered Floer homology of the Seifert surface complement. In particular, we will show that bordered Floer categorifies Donaldson's TQFT description of the Alexander module. This seminar will be an hour long to allow for the GT-MAP seminar at 3 pm.