Georgia Institute of Technology College of Sciences

When studying symplectic 4-manifolds, it is useful to consider Lefschetz fibrations over the 2-sphere due to their one-to-one correspondence uncovered by Freedman and Gompf. Lefschetz fibrations of genera 0 and 1 are well understood, but for genera greater than or equal to 2, much less is known. However, some Lefschetz fibrations with monodromies that respect the hyperelliptic involution of a genus-g surface have stronger properties which make their invariants easier to compute.

I will talk about various notions of equivalence for manifolds and morphisms and the relationships between them. Questions, interruptions, and detours are strongly encouraged! 

The fine curve graph of a surface is a graph whose vertices are essential simple closed curves in the surface and whose edges connect disjoint curves. Following a rich history of hyperbolicity in various graphs based on surfaces, the fine curve was shown to be hyperbolic by Bowden–Hensel–Webb. Given how well-studied the curve graph and the case of “up to isotopy” is, we ask: what about the mysterious part of the fine curve graph not captured by isotopy classes?

The concept of holonomy arises in many areas of mathematics, especially control theory. This concept is also related to the broader program of geometrization of forces in physics. In order to understand holonomy, we need to understand principal (fiber) bundles. In this talk I will explain U(1)-principal bundles by example. This explanation will be from the point-of-view of a geometer, but I will introduce the terminology of control theory. Finally, we will do a holonomy computation for a famous example of Aharonov and Bohm.

In dimension 4, there exist simply connected manifolds which are homeomorphic but not diffeomorphic; the difference between the distinct smooth structures can be localized using corks. Similarly, there exist diffeomorphisms of simply connected 4-manifolds which are topologically but not smoothly isotopic to the identity. In this talk, I will discuss some preliminary results towards an analogous localization of this phenomena using corks for diffeomorphisms. This project is joint work with Slava Krushkal, Anubhav Mukherjee, and Mark Powell.

The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface.  A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3.  We will explain work of Ershov-He and Church-Ershov-Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1.  Time permitting, we will also d

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties (e.g. tightness, fillability, vanishing or non-vanishing of various Floer or symplectic homology classes) of contact structures are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. The case of positive contact surgeries much more subtle.

Braid groups are relatively simple to describe, but they have deep and intricate connections to many different areas of math. We will discuss three specific instances where the braid group on 3 strands arises in geometry and knot theory. In exploring connections between these perspectives, we will take a detour into the world of elliptic curves and their moduli space. As a result, we will see that these three perspectives are actually the same. Time permitting, we will explore generalizations of this to the braid group on n strands for n > 3.

The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface.  A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3.  We will explain work of Ershov—He and Church—Ershov—Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1.  Time permitting, we will also d

This will be an introduction to Legendrian contact homology (LCH), a version of Floer homology that's important in contact topology, for the setting of Legendrian knots in R^3 with the standard contact structure. LCH is the homology of a differential graded algebra that can be defined combinatorially in terms of a diagram for the knot. We'll explore this combinatorial definition, with examples, and discuss some auxiliary invariants derived from LCH. No background about contact manifolds or Legendrian knots will be assumed.