Tag - Symplectic geometry

Angela Wu: On Lagrangian Quasi-Cobordisms

24th April, 2023

A Lagrangian cobordism between Legendrian knots is an important notion in symplectic geometry. Many questions, including basic structural questions about these surfaces are yet unanswered. For instance, while it is known that these cobordisms form a preorder, and that they are not symmetric, it is not known if they form a partial order on Legendrian knots. The idea of a Lagrangian quasi-cobordism was first defined by Sabloff, Vela-Vick, and Wong. Roughly, for two Legendrians of the same rotation number, it is the smooth composition of a sequence of alternatingly ascending and descending Lagrangian cobordisms which start at one knot and ends at the other. This forms a metric monoid on Legendrian knots, with distance given by the minimal genus between any two Legendrian knots. In this talk, I will discuss some new results about Lagrangian quasi-cobordisms, based on some work in progress with Sabloff, Vela-Vick, and Wong.

Amanda Hirschi: Global Kuranishi Charts for Gromov-Witten Moduli Spaces and a Product Formula

21st April, 2023

I will describe the construction of a global Kuranishi chart for moduli spaces of stable pseudoholomorphic maps of any genus and explain how this allows for a straightforward definition of GW invariants. For those not convinced of its usefulness, I will sketch how this can be used to obtain a formula for the GW invariants of a product. This is joint work with Mohan Swaminathan.

Roman Krutowski: Maslov Index Formula in Heegaard Floer Homology

21st April, 2023

The formula introduced by Robert Lipshitz for Heegaard Floer homology is now one of the basic tools for those working with HF homology. The convenience of the formula is due to its combinatorial nature. In the talk, we will discuss the recent combinatorial proof of this formula.

Brayan Ferreira: Gromov Width of Disk Cotangent Bundles of Spheres of Revolution

21st April, 2023

The question of whether a Symplectic manifold embeds into another is central in Symplectic topology. Since Gromov nonsqueezing theorem, it is known that this is a different problem from volume preserving embeddings. Symplectic capacities are invariants that give obstructions to symplectic embeddings. The first example of a symplectic capacity is given by the Gromov width, which measures the biggest ball that can be symplectically embedded into a symplectic manifold. In this talk, we are going to discuss the Gromov width for the example of disk cotangent bundles of spheres of revolution. The main results are for the Zoll cases and for the case of ellipsoids of revolution. The main tools are action angle coordinates (Arnold-Liouville theorem) and ECH capacities.

Joseph Helfer: Exotic contact structures on ℝn

17th April, 2023

Contact homology is a Floer-type invariant for contact manifolds, and is a part of Symplectic Field Theory. One of its first applications was the existence of exotic contact structures on spheres. Originally, contact homology was defined only for closed contact manifolds. We will describe how to extend it to open contact manifolds that are 'convex'. As an application, we prove the existence of (infinitely many) exotic contact structures on ℝ2n+1 for all n > 1.

Pierrick Bousseau: Quivers, Flow Trees and Log Curves

14th April, 2023

Donaldson-Thomas (DT) invariants of a quiver with potential can be expressed in terms of simpler attractor DT invariants by a universal formula. The coefficients in this formula are calculated combinatorially using attractor flow trees. In joint work with Arguz, we prove that these coefficients are genus 0 log Gromov-Witten invariants of d-dimensional toric varieties, where d is the number of vertices of the quiver. This result follows from a log-tropical correspondence theorem which relates (d-2)-dimensional families of tropical curves obtained as universal deformations of attractor flow trees, and rational log curves in toric varieties.

Michael Hutchings: Braid Stability for Periodic Orbits of Area-preserving Surface Diffeomorphisms

3rd April, 2023

Given an area-preserving surface diffeomorphism, what can one say about the topological properties of its periodic orbits? In particular, a finite set of periodic orbits gives rise to a braid in the mapping torus, and one can ask which isotopy classes of braids arise this way. We show that under some nondegeneracy hypotheses, the isotopy classes of braids that arise from finite sets of periodic orbits are stable under Hamiltonian perturbations that are small with respect to the Hofer metric. A corollary is that within a Hamiltonian isotopy class, the topological entropy is lower semicontinuous with respect to the Hofer metric. It is an open question whether analogous statements hold for Reeb orbits on contact three-manifolds.

Georgios Dimitroglou Rizell: A Relative Calabi-Yau Structure for Legendrian Contact Homology

31st March, 2023

The duality long exact sequence relates linearised Legendrian contact homology and cohomology and was originally constructed by Sabloff in the case of Legendrian knots. We show how the duality long exact sequence can be generalised to a relative Calabi-Yau structure, as defined by Brav and Dyckerhoff. We also discuss the generalised notion of the fundamental class and give applications. The structure is established through the acyclicity of a version of Rabinowitz Floer Homology for Legendrian submanifolds with coefficiens in the Chekanov-Eliashberg DGA. This is joint work in progress with Legout.

Yaron Ostrover: Symplectic Barriers

24th March, 2023

In this talk we discuss the existence of a new type of rigidity of symplectic embeddings coming from obligatory intersections with symplectic planes.

Vasily Krylov: On Hikita-Nakajima conjecture for quiver varieties and Slodowy slices

22nd March, 2023

Symplectic duality predicts that symplectic singularities should come in pairs. For example, Nakajima quiver varieties are conjecturally dual to BFN Coulomb branches (of the corresponding quiver theories). Another family of potentially symplectically dual pairs was described recently in the works of Losev, Mason-Brown, and Matvieievskyi: they describe symplectically duals to Słodowy slices to nilpotent orbits.

In this talk, we will discuss the Hikita-Nakajima conjecture that relates the geometry of symplectically dual varieties. It turns out that the conjecture is very likely to hold for quiver varieties (as was predicted by Nakajima) but does not quite hold for Słodowy slices and arbitrary Higgs branches. We will explain certain simplification of this conjecture that may work in general. We will discuss a possible approach toward the proof of this conjecture. The approach is highly based on the ideas of Bellamy, Braverman, Kamnitzer, Losev, Tingley, Webster, Weekes, Yacobi, and their co-authors.

We will illustrate the approach on the examples of ADHM space (for which Hikita-Nakajima conjecture is true as stated) and for certain Słodowy varieties.