Tag - Symplectic 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.

Joé Brendel: Real Lagrangian tori in toric symplectic manifolds

5th June, 2020

In this talk we will be addressing the question whether a given Lagrangian torus in a toric monotone symplectic manifold can be realized as the fixed point set of an anti-symplectic involution (in which case it is called 'real'). In the case of toric fibres, the answer depends on the geometry of the moment polytope of the ambient manifold. In the case of the Chekanov torus, the answer is always no. This can be proved using displacement energy and versal deformations.

Abror Pirnapasov: Reeb orbits that force topological entropy

5th June, 2020

A transverse link in a contact 3-manifold forces topological entropy if every Reeb flow possessing this link as a set of periodic orbits has positive topological entropy. We will explain how cylindrical contact homology on the complement of transverse links can be used to show that certain transverse links force topological entropy. As an application, we show that on every closed contact 3-manifold exists transverse knots that force topological entropy. We also generalize to the category of Reeb flows a beautiful result due to Denvir and Mackay, which says that if a Riemannian metric on the two-dimensional torus has a contractible closed geodesic then its geodesic flow has positive topological entropy.

Alex Oancea: Duality for Rabinowitz-Floer homology

29th May, 2020

I will explain a duality theorem with products in Rabinowitz-Floer homology. This has a bearing on string topology and explains a number of dualities that have been observed in that setting.

Denis Auroux: Mirrors of curves and their Fukaya categories

22nd May, 2020

Homological mirror symmetry predicts that the derived category of coherent sheaves on a curve has a symplectic counterpart as the Fukaya category of a mirror space. However, with the exception of elliptic curves, this mirror is usually a symplectic Landau-Ginzburg model, i.e. a non-compact manifold equipped with the extra data of a 'stop' in its boundary at infinity. Most of the talk will focus on a family of Landau-Ginzburg models which provide mirrors to curves in (C*)2 or in toric surfaces (or more generally to hypersurfaces in toric varieties), and their fiberwise wrapped Fukaya categories (joint work with Mohammed Abouzaid). I will then discuss more a speculative way of constructing mirrors of curves without Landau-Ginzburg models, involving a new flavour of Lagrangian Floer theory in trivalent configurations of Riemann surfaces (joint work with Alexander Efimov and Ludmil Katzarkov).

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.

João Schwarz: Poisson birational equivalence and Coloumb branches of 3d N=4 SUSY gauge theories

14th May, 2020

In this talk we discuss a notion of birational equivalence suitable for Poisson affine varieties: namely, that their function fields are isomorphic as Poisson fields. Some very interesting questions on non-commutative birational geometry, such as the Gelfand-Kirillov Conjecture, make perfect sense in the quasi-classical limit, and naturally leads one to consider the Poisson birational class of the algebras they quantize. In this setting, we study the behaviour of Poisson birational equivalence on the quasi-classical limit of rings of differential operators. With this idea we solve a Poisson analogue of Noether's Problem, introduced by Julie Baudry and François Dumas, in a constructive fashion, for essentially all finite symplectic reflection groups. As applications of our method, we show the Poisson rationality of the Generalized Calogero-Moser spaces, introduced by Etingof and Ginzburg in 2002, and surprisngly for this author, all Coloumb branches of 3d, N=4 SUSY gauge theories - an important object in mathematical physics recently given a rigorous formulation by Nakajima in 2015, and later Nakajima, Braverman, Finkelberg in 2016.

Marco Mazzucchelli: Spectral characterizations of Besse and Zoll Reeb flows

8th May, 2020

In this talk, I will address a geometric inverse problem from contact geometry: is it possible to recognize whether all orbits of a given Reeb flow are closed from the knowledge of the action spectrum? Borrowing the terminology from Riemannian geometry, Reeb flows all of whose orbits are closed are sometimes called Besse, and Besse Reeb flows all of whose orbits have the same minimal period are sometimes called Zoll. In the talk I will summarize recent results on this inverse problem in a few settings: geodesic flows (joint work with Stefan Suhr), closed contact 3-manifolds (joint work with Daniel Cristofaro-Gardiner), convex contact spheres and, more generally, restricted contact type hypersurfaces of symplectic vector spaces (joint work with Viktor Ginzburg and Basak Gürel). I will also mention a few conjectures and open problems.

Lisa Traynor: The Geography of Immersed Lagrangian Fillings of Legendrian Submanifolds

24th April, 2020

Given a smooth knot K in the 3-sphere, a classic question in knot theory is: What surfaces in the 4-ball have boundary equal to K? One can also consider immersed surfaces and ask a 'geography' question: What combinations of genus and double points can be realized by surfaces with boundary equal to K? I will discuss symplectic analogues of these questions: Given a Legendrian knot, what Lagrangian surfaces can it bound? What immersed Lagrangian surfaces can it bound? These Lagrangian surfaces are commonly called Lagrangian fillings of the Legendrian knot and are more rigid than their topological counterpart. In particular, while any smooth knot bounds an infinite number of topologically distinct surfaces, there are classical and non-classical obstructions to the existence of Lagrangian fillings of Legendrian knots. Specifically, a polynomial associated to the Legendrian boundary through the technique of generating families can show that there is no compatible embedded Lagrangian filling. Immersed Lagrangian fillings are more flexible, and I will describe how this polynomial associated to the Legendrian boundary forbids particular combinations of genus and double points in immersed Lagrangian fillings. In addition, I will describe some constructions of immersed fillings that allow us to completely answer the Lagrangian geography question for some Legendrian knots.

Nicholas Wilkins: Equivariant quantum operations and relations between them

17th April, 2020

There is growing interest in looking at operations on quantum cohomology that take into account symmetries in the holomorphic spheres (such as the quantum Steenrod powers, using a ℤ/p-symmetry). In order to prove relations between them, one needs to generalise this to include equivariant operations with more marked points, varying domains and different symmetry groups. We will look at the general method of construction of these operations, as well as two distinct examples of relations between them.