Georgia Institute of Technology College of Sciences

Real Heegaard Floer homology is a new invariant of branched double covers, introduced by Gary Guth and Ciprian Manolescu, and inspired by work of Jiakai Li and others in Seiberg-Witten theory. After sketching their construction, we will describe an extension of the "hat" variant to 3-manifolds with boundary, and the algorithm this gives to compute it when the fixed set is connected. We will end with some open questions.

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.

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.