Georgia Institute of Technology College of Sciences

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.

Vassiliev knot invariants, or finite-type invariants, are a broad class of knot invariants resulting from extending usual invariants to knots with transverse double points. We will show that the Conway and Jones polynomials are fully described by Vassiliev invariants. We will discuss the fundamental theorem of Vassiliev invariants, relating them to the algebra of chord diagrams and weight systems. Time permitting, we will also discuss the Kontsevich integral, the universal Vassiliev invariant.

Legendrian knots are an important kind of knot in contact topology. One of their invariants,  the Thurston-Bennequin number, has an upper bound for any given knot type, called max-tb. Using convex surface theory, we will compute the max-tb of positive torus knots and show that two max-tb positive torus knots are Legendrian isotopic. If time permits, we will show that any non max-tb positive torus knot is obtained from the unique max-tb positive torus knot by a sequence of stabilizations. 

Mapping class groups of surfaces in general have cohomology that is hard to compute. Meanwhile, within something called the cohomologically-stable range, a family of characteristic classes called the MMM classes (of surface bundles) is enough to generate this cohomology and thus plays an important role for understanding both the mapping class group and surface bundles. Moreover, constructing the so-called Atiyah-Kodaira manifold provides the setting to prove that these MMM classes are non-trivial.

In this talk, we will give background on Lefschetz fibrations and their relationship to symplectic 4-manifolds. We will then discuss results on their fundamental groups. Genus-2 Lefschetz fibrations are of particular interest because of how much we know and don't know about them. We will see what fundamental groups a genus-2 Lefschetz fibration can have and what questions someone might ask when studying these objects.

Morse theory analyzes the topology of a smooth manifold by studying the behavior of its real-valued functions. From this, one obtains a well-behaved homology theory which provides further information about the manifold and places constraints on the smooth functions it admits. This idea has proven to be useful in approaching topological problems, playing an essential role in Smale's solution to the generalized Poincare conjecture in dimensions greater than 4.

This talk will include background information on contact structures and open book decompositions of 3-manifolds and the relationship between them. I will state the necessary definitions and include a number of concrete examples. I will also review some convex surface theory, which is the tool used in the main talk to investigate the contact structure – open book relationship.