Tag - Symplectic geometry

Igor Uljarević: Contact non-squeezing via selective symplectic homology

14th October, 2022

I will introduce a new version of symplectic homology that resembles the relative symplectic homology and that is related to the symplectic homology of a Liouville sector. This version, called selective symplectic homology, is associated with a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of the Floer homology groups for Hamiltonians whose slopes tend to infinity on the open subset but remain close to 0 and positive on the rest of the boundary. As an application, I will prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with infinite dimensional symplectic homology: there exists a smoothly embedded closed ball in such a sphere that cannot be made arbitrarily small by a contact isotopy. These homotopy spheres include examples that are diffeomorphic to standard spheres and whose contact structures are homotopic to standard contact structures.

Julian Chaidez: The Ruelle invariant and convexity in higher dimensions

24th June, 2022

I will explain how to construct the Ruelle invariant of a symplectic cocycle over an arbitrary measure preserving flow. I will provide examples and computations in the case of Hamiltonian flows and Reeb flows (in particular, for toric domains). As an application of this invariant, I will construct toric examples of dynamically convex domains that are not symplectomorphic to convex ones in any dimension.

Yoel Groman: Locality and deformations in relative symplectic cohomology

17th June, 2022

Relative symplectic cohomology is a Floer theoretic invariant associated with compact subsets K of a closed or geometrically bounded symplectic manifold M. The motivation for studying it is that it is often possible to reduce the study of global Floer theory of M to the Floer theory of a handful of local models covering M which one hopes will be easier to compute (Varolgunes' spectral sequence). As an example, it is expected that at least in the setting of the Gross-Siebert program, the mirror can be pieced together from the relative symplectic cohomologies of neighborhoods of fibers of an SYZ fibration (singular or not). However, even when K is a well understood model, such as the Weinstein neighborhood of a Lagrangian torus, the construction of relative SH is rather unwieldy. In particular, it is not entirely obvious how to relate the symplectic cohomology of K relative to M with Floer theoretic invariants intrinsic to K. I will discuss a number of results, most of them in preparation, which aim to alleviate this difficulty in the setting Lagrangian torus fibrations with singularities.

Guangbo Xu: Integer-valued Gromov-Witten type invariants

3rd June, 2022

Gromov-Witten invariants for a general target are rational-valued but not necessarily integer-valued. This is due to the contribution of curves with nontrivial automorphism groups. In 1997 Fukaya and Ono proposed a new method in symplectic geometry which can count curves with a trivial automorphism group. While ordinary Gromov-Witten invariants only use the orientation on the moduli spaces, this integer-valued counts are supposed to also use the (stable) complex structure on the moduli spaces. In this talk, I will present the recent joint work with Shaoyun Bai in which we rigorously defined the integer-valued Gromov-Witten type invariants in genus zero for a symplectic manifold.

Miguel Pereira: The Lagrangian capacity of toric domains

27th May, 2022

In this talk, I will state a conjecture giving a formula for the Lagrangian capacity of a convex or concave toric domain. First, I will explain a proof of the conjecture in the case where the toric domain is convex and 4-dimensional, using the Gutt-Hutchings capacities as well as the McDuff-Siegel capacities. Second, I will explain a proof of the conjecture in full generality, but assuming the existence of a suitable virtual perturbation scheme which defines the curve counts of linearized contact homology. This second proof makes use of Siegel's higher symplectic capacities.

Maksim Stokić: C0 contact geometry of isotropic submanifolds

27th May, 2022

Homeomorphism is called contact if it can be written as C0-limit of contactomorphisms. The contact version of Eliashberg-Gromov rigidity theorem states that smooth contact homeomorphisms preserve contact structure. Submanifold L of a contact manifold (Y, ξ) is called isotropic if ξ|TL =0. Isotropic submanifolds of maximal dimension are called Legendrian, otherwise we call them subcritical isotropic. In this talk, we will try to answer whether the isotropic property is preserved by contact homeomorphisms. It is expected that subcritical isotropic submanifolds are flexible, while we expect that Legendrians are rigid. We show that subcritical isotropic curves are flexible, and we give a new proof of the rigidity of Legendrians in dimension 3. Moreover, we provide a certain type of rigidity of Legendrians in higher dimensions.

Daniel Rudolf: Viterbo’s conjecture for Lagrangian products in ℝ4

27th May, 2022

We show that Viterbo's conjecture (for the EHZ-capacity) for convex Lagrangian products in ℝ4 holds for all Lagrangian products (any trapezoid in ℝ2) x (any convex body in ℝ2). Moreover, we classify all equality cases of Viterbo’s conjecture within this configuration and show which of them are symplectomorphic to a Euclidean ball. As by-product, we conclude sharp systolic Minkowski billiard inequalities for geometries which have trapezoids as unit balls. Finally, we show that the flows associated to the above mentioned equality cases (which are polytopes) satisfy a weak Zoll property, namely, that every characteristic that is almost everywhere away from lower-dimensional faces is closed, runs over exactly 8 facets, and minimizes the action.

Claude Viterbo: γ-support, γ-coisotropic subsets and application

20th May, 2022

To an element in the completion of the set of Lagrangians for the spectral distance we associate a support. We show that such a support is γ-coisotropic (a notion we shall define in the talk) and we shall give examples and counterexamples of γ-coisotorpic sets that can be (or cannot be) γ-supports. Finally, we give some applications of these notions to singular support of sheaves (joint work with S. Guillermou) and dissipative dynamics, allowing us to extend the notion of Birkhoff attractor (joint with V. Humilière).

Giulia Saccà: Moduli spaces on K3 categories are Irreducible Symplectic Varieties

7th May, 2022

Recent developments by Druel, Greb-Guenancia-Kebekus, Horing-Peternell have led to the formulation of a decomposition theorem for singular (klt) projective varieties with numerical trivial canonical class. Irreducible symplectic varieties are one the building blocks provided by this theorem, and the singular analogue of irreducible hyper-Kahler manifolds. In this talk I will show that moduli spaces of Bridgeland stable objects on the Kuznetsov component of a cubic fourfold with respect to a generic stability condition are always projective irreducible symplectic varieties. I will rely on the recent work of Bayer-Lahoz-Macri-Neuer-Perry-Stellari, which, ending a long series of results by several authors, proved the analogue statement in the smooth case.

Kevin Ruck: Tate Homology and powered Flybys

6th May, 2022

In this talk, I want to show that in the planar circular restricted three body problem there are infinitely many symmetric consecutive collision orbits for all energies below the first critical energy value. By using the Levi-Civita regularization we will be able to distinguish between two different orientations of these orbits and prove the above claim for both of them separately. In the first part of the talk, I want to explain the motivation behind this result, especially its connection to powered Flybys. Afterward I will introduce the main technical tools, one needs to prove the above statement, like Lagrangian Rabinowitz Floer Homology and its G-equivariant version. To be able to effectively calculate this G-equivariant Lagrangian RFH, we will relate it to the Tate homology of the group G. With this tool at hand, we will then finally be able to prove that there are infinitely many consecutive collision orbits all facing in a specific direction.