Georgia Institute of Technology College of Sciences

I'll report on an ongoing project, partly joint work with J. Hillman, aimed at finding criteria for the existence of sections on a given Lefschetz fibration over a surface. We will start by presenting a nice algebraic criterion for the existence of sections in a surface bundle and explain what goes wrong if we try to apply it to the more general Lefschetz fibration case. The question of when a nullhomotopic loop in the boundary of a Lefschetz fibration over the disk can be extended to a section over the whole disk is one such subtle issue.

Abstract TBA

The Fox trapezoidal conjecture is a longstanding open problem about the coefficients of the Alexander polynomial of alternating links. In this talk, we will discuss recent work which settled this conjecture for “special alternating links”. The first tool is a graph theoretic model of the Alexander polynomial of an alternating link discovered by Crowell in 1959. The second is the theory of Lorentzian polynomials, developed by Brändén and Huh in 2019 and a key part of Huh’s Fields medal work.

In order to distinguish Legendrians with the same classical invariants, Chekanov and Eliashberg separately defined the Chekanov-Eliashberg DGA. Chekanov further defined a linearized version. Ekholm, Honda, and Kalman showed an exact Lagrangian cobordism between two Legendrians induces a DGA map on their respective DGAs. We show how to adapt this map to the linearized version. Time permitting, we will use this map to obstruct invertible concordances between negative twist knots. This is joint work with Sierra Knavel.

Symplectic manifolds exhibit curious behaviour at the interface of rigidity and flexibility. A non-squeezing phenomenon discovered by Gromov in the 1980s was the first manifestation of this. Since then, extensive research has been carried out into when standard symplectic shapes embed inside another -- it turns out that even when volume obstructions vanish, sometimes they cannot. A mysterious connection to Markov numbers, a generalization of the Fibonacci numbers, and an infinite staircase, is exhibited in the study of embeddings of ellipsoids into balls.

On a closed, simply-connected, symplectic 4-manifold, the Dehn–Seidel twists on Lagrangian spheres and their products provide all known examples of non-trivial elements in the symplectic mapping class group. However, little is known in general about the relations that may hold among Dehn–Seidel twists. 

Giroux torsion is an important class of contact structures on a neighborhood of a torus, which is known to obstruct symplectic fillability. Ghiggini conjectured that half Giroux torsion along a separating torus always results in a vanishing Heegaard Floer contact invariant hence also obstructs fillability. In this talk, we present a counterexample to that conjecture. Our main tool is a bordered contact invariant, which enables efficient computation of the contact invariant.

Satellite operations are one of the most basic operations in knot theory. Many researchers have studied the behavior of knot Floer homology under satellite operations. Most of these results use Lipshitz, Ozsvath and Thurston's bordered Heegaard Floer theory. In this talk, we discuss a new technique for studying these operators, and we apply this technique to a family of operators called L-space operators. Using this theory, we are able in many cases to give a simple formula for the behavior of the concordance invariant tau under such operators.

It is a classical problem to study whether the h-principle holds for certain classes of maximally non-integrable distributions. The most studied case is that of contact structures, where there is a rich interplay between flexibility and rigidity, exemplified by the overtwisted vs tight dichotomy. For other types of maximally non-integrable distributions, no examples of rigidity are currently known.