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.

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.

Giroux and Pardon conjectured that a Lagrangian L in a Weinstein manifold W is regular (that is, compatible with the Weinstein structure in a natural sense) if there is a Lefschetz fibration p: W \to \C such that p(L) is a ray. In this talk, I will discuss forthcoming joint work with A. Roy and L. Wang, which establishes this conjecture. As an application of the proof, we show how all fillings of the rainbow closures of a positive braid can be described by manipulations of arcs in the base of an appropriate Lefschetz fibration.

Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.

This thesis adopts the immersed-curve perspective to analyze the knot Floer complexes of (1,1)-satellite knots. The main idea is to encode the chain model construction through what we call a planar (1,1)-pairing. This combinatorial and geometric object captures the interaction between the companion and the pattern via the geometry of immersed and embedded curves on a torus (or its planar lift).

We discuss the proof of the following Theorem

Assume $E$ is a $C^{p}$ real Banach manifold, and $f:E\circlearrowleft$, $f\circ f=f$ is a $C^{p}$ retraction, where $1\leq p\leq +\infty$. Then the retract $f(E)$ is a $C^{p}$ sub Banach manifold of $E$.