Tag - Contact geometry

Morgan Weiler: Infinite staircases of symplectic embeddings of ellipsoids into Hirzebruch surfaces

5th June, 2020

Gromov nonsqueezing tells us that symplectic embeddings are governed by more complex obstructions than volume. In particular, in 2012, McDuff-Schlenk computed the embedding capacity function of the ball, whose value at a is the size of the smallest four-dimensional ball into which the ellipsoid E(1,a) symplectically embeds. They found that it contains an 'infinite staircase' of piecewise-linear sections accumulating from below to the golden ratio to the fourth power. However, infinite staircases seem to be rare for more general targets. Work of Cristofaro-Gardiner-Holm-Mandini-Pires suggests that, up to scaling, there are only finitely many rational symplectic toric manifolds whose embedding capacity functions contain infinite staircases, while Usher has found infinitely many irrational polydisks with infinite staircases. Using ECH capacities in conjunction with the methods of McDuff-Schlenk, we will explain how we have found several infinite families of Hirzebruch surfaces whose embedding capacity functions we expect to contain an infinite staircase. Many of these staircases are 'descending' rather than 'ascending'. This is joint work with Maria Bertozzi, Tara Holm, Emily Maw, Dusa McDuff, Grace Mwakyoma, and Ana Rita Pires.

Jo Nelson: Reflections on Cylindrical Contact Homology

15th May, 2020

This talk beings with a light introduction, including some historical anecdotes to motivate the development of this Floer theoretic machinery for contact manifolds some 25 years ago. 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 explain the need for an obstruction bundle gluing correction term in the expression of the differential in the presence of contractible Reeb orbits, which is essential even in the simple example of an ellipsoid. I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our constructions.

Lenny Ng: Recent developments in knot contact homology

11th December, 2017

Knot contact homology is a knot invariant derived from counting holomorphic curves with boundary on the Legendrian conormal to a knot. I will discuss some new developments around the subject, including an enhancement that completely determines the knot (joint work with Tobias Ekholm and Vivek Shende) and recent progress in the circle of ideas connecting knot contact homology, recurrence relations for coloured HOMFLY polynomials, and topological strings (joint work in progress with Tobias Ekholm).

Sheng-Fu Chiu: Sheaves and contact non-squeezing in ℝ2n × S1

13th April, 2017

In this talk I will introduce a way to associate a triangulated category of sheaves with a domain of ℝ2n×S1. The cohomological information on the category side helps to detect the contact non-squeezing property of the domain on the topology side.

Dusa McDuff: The stabilized symplectic embedding problem

30th March, 2017

I will discuss some recent work (mostly joint with Dan Cristofaro-Gardiner and Richard Hind) on the stabilized symplectic embedding problem for ellipsoids into balls. The main tools come from embedded contact homology.

Kei Irie: C∞ closing lemma for three-dimensional Reeb flows via embedded contact homology

16th February, 2017

Cr closing lemma is an important statement in the theory of dynamical systems, which implies that for a Cr generic system the union of periodic orbits is dense in the nonwondering domain. C1 closing lemma is proved in many classes of dynamical systems, however Cr closing lemma with r > 1 is proved only for few cases. In this talk, I'll prove C∞ closing lemma for Reeb flows on closed contact three-manifolds. The proof uses recent developments in quantitative aspects of embedded contact homology (ECH). In particular, the key ingredient of the proof is a result by Cristofaro-Gardiner, Hutchings and Ramos, which claims that the asymptotics of ECH spectral invariants recover the volume of a contact manifold. Applications to closed geodesics on Riemannian two-manifolds and Hamiltonian diffeomorphisms of symplectic two-manifolds (joint work with M. Asaoka) will be also presented.

Joshua Sabloff: Length and width of Lagrangian cobordisms

11th October, 2016

In this talk, I will discuss two measurements of Lagrangian cobordisms between Legendrian submanifolds in symplectizations: their length and their relative Gromov width. The Gromov width, in particular, is a fundamental global invariant of symplectic manifolds, and a relative version of that width helps understand the geometry of Lagrangian submanifolds of a symplectic manifold. Lower bounds on both the length and the width may be produced by explicit constructions; this talk will concentrate on upper bounds that arise from a filtered version of Legendrian contact homology, a Floer-type invariant.

Cagatay Kutluhan: A Heegaard Floer analogue of algebraic torsion

21st April, 2016

The dichotomy between overtwisted and tight contact structures has been central to the classification of contact structures in dimension 3. Ozsvath-Szabo's contact invariant in Heegaard Floer homology proved to be an efficient tool to distinguish tight contact structures from overtwisted ones. In this talk, I will motivate, define, and discuss some properties of a refinement of the contact invariant in Heegaard Floer homology.

Ana Rita Pires: Symplectic embeddings and infinite staircases

15th April, 2016

McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase determined by the odd index Fibonacci numbers. Infinite staircases have also been shown to exist in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid E(2,3). This talk describes joint work with Dan Cristofaro-Gardiner, Tara Holm, and Alessia Mandini, in which we use ECH capacities to show that infinite staircases exist for these and a few other target manifolds. I will also explain why we conjecture that these are the only such twelve.