Symplectic Dynamics/Geometry Seminar

Inscribing Rectangles in Jordan Loops

Rich Schwartz
Brown University
October 14, 2019

I'll show a graphical user interface I wrote which explores the problem of inscribing rectangles in Jordan loops.  The motivation behind this is the notorious Square Peg Conjecture of Toeplitz, from 1911.

I did not manage to solve this problem, but I did get the result that at most 4 points of any Jordan loop are vertices of inscribed  rectangles. I will sketch a proof of this result, mostly through visual demos, and also I will explain two other theorems about inscribed rectangles which at least bear a resemblance to theorems in symplectic geometry.

Bourgeois contact structures: tightness, fillability and applications.

Agustin Moreno
University of Augsburg
October 7, 2019
Starting from a contact manifold and a supporting open book decomposition, an explicit construction by Bourgeois provides a contact structure in the product of the original manifold with the two-torus. In this talk, we will discuss recent results concerning rigidity and fillability properties of these contact manifolds. For instance, it turns out that Bourgeois contact structures are, in dimension 5, always tight, independent on the rigid/flexible classification of the original contact manifold.

Equivariant and nonequivariant contact homology

Jo Nelson
Rice University
March 20, 2019

I will discuss joint work with Hutchings which constructs nonequivariant and a family floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over Z and capture interesting torsion information. I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our construction.

Minimal Sets and Properties of Feral Pseudoholomorphic Curves

Joel Fish
University of Massachusetts Boston
March 18, 2019

I will discuss some current joint work with Helmut Hofer, in which we define and establish properties of a new class of pseudoholomorhic curves (feral J-curves) to study certain divergence free flows in dimension three. In particular, we show that if H is a smooth, proper, Hamiltonian on R^4, then no non-empty regular energy level of H is minimal. That is, the flow of the associated Hamiltonian vector field has a trajectory which is not dense.

Gysin sequences and cohomology ring of symplectic fillings

Zhengyi Zhou
Member, School of Mathematics
March 4, 2019

It is conjectured that contact manifolds admitting flexible fillings have unique exact fillings. In this talk, I will show that exact fillings (with vanishing first Chern class) of a flexibly fillable contact (2n-1)-manifold share the same product structure on cohomology if one of the multipliers is of even degree smaller than n-1. The main argument uses Gysin sequences from symplectic cohomology twisted by sphere bundles.

Higher symplectic capacities

Kyler Siegel
Columbia University
February 25, 2019
I will describe a new family of symplectic capacities defined using rational symplectic field theory.
These capacities are defined in every dimension and give state of the art obstructions for various "stabilized" symplectic embedding problems such as one ellipsoid into another. They can also be described via symplectic cohomology and are related to counting pseudoholomorphic curves with tangency conditions. I will explain the basic idea of the construction and then give some computations, structural results, and applications.

Barcodes and $C^0$ symplectic topology

Sobhan Seyfaddini
ENS Paris
December 17, 2018
Hamiltonian homeomorphisms are those homeomorphisms of a symplectic manifold which can be written as uniform limits of Hamiltonian diffeomorphisms. One difficulty in studying Hamiltonian homeomorphisms (particularly in dimensions greater than two) has been that we possess fewer tools for studying them. For example, (filtered) Floer homology, which has been a very effective tool for studying Hamiltonian diffeomorphisms, is not well-defined for homeomorphisms.