Georgia Institute of Technology College of Sciences

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.

For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids.

Abstract: Fundamental to our understanding of Teichm\"uller space T(S) of a closed oriented genus $g \geq 2$ surface S are two different perspectives: one as connected  component in the  PSL(2,\R) character variety  \chi(\pi_1S, PSL(2,\R)) and one as the moduli space of marked hyperbolic structures on S. The latter can be thought of as a moduli space of (PSL(2,\R), \H^2) -structures. The G-Hitchin component, denoted Hit(S,G), for G a split real simple Lie group, is a connected component in \chi(\pi_1S, G) that is a higher rank generalization of T(S).

A sequence of remarkable results in recent decades have shown that for a surface group H there are many Lie groups G and connected components C of Hom(H,G) consisting of discrete and faithful representations. These are known as higher Teichmüller spaces. With two exceptions, all known constructions of higher Teichmüller spaces work only for surface groups. This is an expository talk on the remarkable paper Convexes Divisibles III (Benoist ‘05), in which the first construction of higher Teichmüller spaces that works for some non-surface-groups was discovered.

 This talk is a summary of a summary. We will be going over Jen Hom's 2024 Levi L. Conant Prize Winning Article "Getting a handle on the Conway knot," which discusses Lisa Piccirillo's renowned 2020 paper proving the Conway knot is not slice. In this presentation, we will go over what it means for a knot to be slice, past attempts to classify the Conway knot with knot invariants, and Piccirillo's approach of constructing a knot with the same knot trace as the Conway knot.