Georgia Institute of Technology College of Sciences

This talk will have two parts.  The first half will describe how to construct symplectic structures on trisected 4-manifolds. This construction is inspired by projective complex geometry and completely characterizes symplectic 4-manifolds among all smooth 4-manifolds.  The second half will address a curious phenomenon: symplectic 4-manifolds appear to not admit any interesting connected sum decompositions.  One potential explanation is that every embedded 3-sphere can be made contact-type.

A link of an isolated complex surface singularity is the intersection of the surface with a small sphere centered at the singular point. The link is a smooth 3-manifold that carries a natural contact structure (given by complex tangencies); one might then want to study its symplectic or Stein fillings. A special family of Stein fillings, called Milnor fillings, can be obtained by smoothing the singular point of the original complex surface.

In the Instanton and Heegaard Floer theories, a nearly fibered knot is one for which the top grading has rank 2. Sivek-Baldwin and Li-Ye showed that the guts (ie. the reduced sutured manifold complement) of a minimal genus Seifert surface of a nearly fibered knot has of one of three simple types.We show that nearly fibered knots with guts of two of these types have handle number 2 while those with guts of the third type have handle number 4.

Braid groups belong to a broad class of groups known as Artin groups, which are defined by presentations of a particular form and have played a major role in geometric group theory and low-dimensional topology in recent years. These groups fall into two classes, finite-type and infinte-type Artin groups. The former come equipped with a powerful combinatorial structure, known as a Garside structure, while the latter are much less understood and present many challenges.

Disks are nice for many reasons. In this casual talk, I will try to convince you that it's even nicer than you think by presenting the Alexander's lemma. Just like in algebraic topology, we are going to rely on disks heavily to understand mapping class groups of surfaces. The particular method is called the Alexander's method. Twice the Alexander, twice the fun! No background in mapping class group is required.

Morse theory and Morse homology together give a method for understanding how the topology of a smooth manifold changes with respect to a filtration of the manifold given by sub-level sets. The Morse homology of a smooth manifold can be expressed using an algebraic object called a persistence module. A persistence module is a module graded by real numbers, and in this setup the grading on the module corresponds to the aforementioned filtration on the smooth manifold.

We will talk a little about realizing automorphisms of a free group as graph maps and how to use Stallings folds to study them.

How good of an invariant is the Jones polynomial? The question is closely tied to studying braid group representations since the Jones polynomial can be defined as a (normalized) trace of a braid group representation.