Georgia Institute of Technology College of Sciences

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. In other cases ingenious constructions such as folding have been invented to find embeddings. In recent work with Hutchings, Weiler, and Yao, I studied the embedding problem for four-dimensional symplectic shapes in conjunction with the question of existence of symplectic surfaces (called anchors) inside the embedding region. In this talk I will survey some of the results and time permitting, discuss the ideas behind the proof, and applications of these techniques to understanding isotopy classes of embeddings.

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. This formula generalizes a large number of existing formulas for the behavior of tau under satellite operations (such as cabling, 1-bridge braids and generalized Mazur patterns), and also has a number of topological applications. This is joint work with Daren Chen and Hugo Zhou.

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. 

I will discuss the following result: on a symplectic K3 surface, the squared Dehn--Seidel twists on Lagrangian spheres with distinct fundamental classes are algebraically independent in the abelianization of the (smoothly-trivial) symplectic mapping class group. In a particular case, this establishes an abelianized form of a Conjecture by Seidel and Thomas on the faithfulness of certain Braid group representations in the symplectic mapping class group of K3 surfaces. The proof makes use of Seiberg--Witten gauge theory for families of symplectic 4-manifolds.

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.

In this talk I will discuss rigidity phenomena for fat distributions, which can be viewed as higher corank generalizations of contact structures. These admit natural symplectizations and contactizations. I will introduce a natural class of submanifolds in fat manifolds, called prelegendrians, which admit canonical Legendrian lifts to the contactization. The main result of the talk is that these submanifolds exhibit rigidity: in the “standard corank-2 fat manifold” there exists an infinite family of prelegendrian tori, all of them formally equivalent but pairwise not prelegendrian isotopic. In other words, the h-principle fails for prelegendrians. The talk is based on joint work with Álvaro del Pino and Wei Zhou.