Georgia Institute of Technology College of Sciences

The Poincaré metric on the unit disc $\mathbb{D} \subset \mathbb{C}$, known for its invariance under all biholomorphisms (bijective holomorphic maps) of $\mathbb{D}$, is one of the most fundamental Riemannian metrics in differential geometry.

In this presentation, we will first introduce the Bergman metric on a bounded domain in $\mathbb{C}^n$, which can be viewed as a generalization of the Poincaré metric. We will then explore some key theorems that illustrate how the curvature of the Bergman metric characterizes bounded domains in $\mathbb{C}^n$ and more generally, complex manifolds. Finally, I will discuss my recent work related to these concepts. 

There are many bizarre examples in the world of smooth 4-manifolds. We will describe some of these examples and discuss a tool from gauge theory that may be used to compute some of the invariants that distinguish these weird examples.

In 1976, Thurston decidedly showed that symplectic geometry and Kähler geometry were strictly distinct by providing the first example of a compact symplectic manifold which is not symplectomorphic to any Kähler manifold. Since this example, first studied by Kodaira, much work has been done in explicating the difference between algebraic manifolds such as affine and projective varieties, complex manifolds such as Stein and Kähler manifolds, and general symplectic manifolds. By building on work first outlined by Seidel, McLean has produced numerous examples of non-affine symplectic manifolds, symplectic manifolds which are not symplectomorphic to any affine variety. McLean approached this problem via analysis of the growth rate of symplectic homology for affine varieties. Every affine variety admits a compactification to a projective variety by a normal crossing divisor. Using this fact, McLean is able to show that the symplectic homology of any affine variety must have a well-controlled growth rate.

We add a bit of subtlety to this already mysterious relationship by providing a particularly interesting example of a non-affine symplectic 4-manifold which admits many normal crossing divisor compactifications. Because of the existence of these nice compactifications, one cannot use growth rate techniques to obstruct our example from being affine and thus cannot apply the work of McLean and Seidel. Our approach to proving this results goes by considering the collection of all symplectic normal crossing divisor compactifications of a particular Liouville manifold  given as a submanifold of the Kodaira-Thurston example . By studying the local geometry of a large collection of symplectic normal crossing divisors, we are able to make several topological conclusions about this collection for  as well as for more general Liouville manifolds  which admit similar compactifications. Our results suggest that a more subtle obstruction must exist for non-affine manifolds. If time permits, we will discuss several structural conclusions one may reach about the collection of divisor compactifications for a more general class of Liouville 4-manifolds.

A fundamental result in 3-dimensional contact topology due to Ding-Geiges tells us that any contact 3-manifold can be obtained via doing a surgery on a Legendrian link in the standard contact 3-sphere. So it's natural to ask how simple or complicated a surgery diagram could be for a given contact manifold? Contact surgery number is a measure of  this complexity. In this talk, I will discuss this notion of complexity along with some examples. This is joint work with Marc Kegel.