Symplectix 6 décembre 2024

 Location: IHP, room 201

The talks are broadcasted via Zoom:
https://zoom.us/j/89445897442?pwd=QmVZWHBiM2Nwb09yUVJERnYrSHJrUT09

 

10:45 Emmanuel Opshtein (Université de Strasbourg)  
Symplectic/contact rigidity of some Lagrangian/Legendrian skeleta in dimension 4.
Abstract: In the simplest framework of symplectic manifolds with rational symplectic class, symplectic polarizations are  smooth symplectic hypersurfaces Poincaré-Dual to a multiple of the symplectic class. Their complements retract to some skeleta, which are quite often Lagrangian CW-complexes. These notions were  introduced by Biran and he exhibited symplectic rigidity properties of these skeleta. In later work, I generalized the notion of symplectic polarizations to any closed symplectic manifold with a view towards effective constructions of symplectic embeddings.
    In the present talk, I will explain further generalization of these notions to the Liouville setting in dimension 4 and how it leads to new interesting results on the side of the symplectic rigidity of Lagrangian skeleta and the contact rigidiy of their Legendrian boundaries (for instance in terms of interlinking in Entov-Polterovich’s words). The talk will focus on examples.   
    This is a joint work with Felix Schlenk.

14:00 (Joint session with the Enumerative Geometry Seminar)
Sebastian Haney (Columbia University)  
Open enumerative mirror symmetry for lines in the mirror quintic.
Abstract:  One of the earliest achievements of mirror symmetry was the prediction of genus zero Gromov-Witten invariants for the quintic threefold in terms of period integrals on the mirror. Analogous predictions for open Gromov-Witten invariants in closed Calabi-Yau threefolds can be formulated in terms of relative period integrals on the mirror, which govern extensions of variations of Hodge structure. I will discuss work in which I construct an immersed Lagrangian in the quintic which supports a family of objects in the Fukaya category mirror to vector bundles on lines in the mirror quintic, and deduce its open Gromov-Witten invariants from homological mirror symmetry. The domain of this Lagrangian immersion is a closed 3-manifold obtained by gluing together two copies of a cusped hyperbolic 3-manifold. The open Gromov-Witten invariants of the Lagrangian are irrational numbers valued in the invariant trace field of the hyperbolic pieces.
 

15:45 Marco Robalo (Sorbonne Université)
Gluing Donaldson-Thomas invariants.
Abstract: Donaldson-Thomas invariants appear naturally in context of (algebraic) lagrangian intersections. Mirror symmetry led Kapustin-Rozansky to conjecture the existence of a particular invariant, related to the categories of matrix factorizations appearing in the B-model.
    In this talk, I will explain a joint work with the B.Hennion (Orsay) and J. Holstein (Hamburg) constructing these conjectural categorical invariants by a procedure that glues what happens in the simplest example, namely, when the lagrangian intersection is given by the critical points of a function.

Next symplectix:

10/01 (Şavk, Shende, Yu), 7/03, 4/04

Other symplectic activity in Paris (and in France):

- CAST 2025 Workshop (6-8 February in Grenoble)
- Séminaire Nantes-Orsay
- Symplectic Zoominar (every Fridays at 15:15 (except Symplectix' Fridays) Paris time)

Symplectix 8 novembre 2024

 Location: IHP, room 201

The talks are broadcasted via Zoom:
https://zoom.us/j/89445897442?pwd=QmVZWHBiM2Nwb09yUVJERnYrSHJrUT09

 

10:45 Marcelo Atallah (Shefield)  
The number of periodic points of surface symplectomorphisms.
Abstract: A celebrated result of Franks shows that a Hamiltonian diffeomorphism of the sphere with more than two fixed points must have infinitely many periodic points. We present a symplectic variant of this phenomenon for symplectomorphisms of surfaces of higher genus that are isotopic to the identity; it implies an upper bound for the Floer-homological count of the number of fixed points of a symplectomorphism with finitely many periodic points. From a higher dimensional viewpoint, this can be understood as evidence for a non-Hamiltonian variant of Shelukhin’s result on the Hofer-Zehnder conjecture. Furthermore, we discuss the construction of a symplectic flow on a surface of any positive genus having a single fixed point and no other periodic orbits. This is joint work with Marta Batoréo and Brayan Ferreira.

14:00 Rima Chatterjee (Köln)  
Classification problem of Legendrian knots vs links.
Abstract:  A knot in a contact manifold is Legendrian if it is everywhere tangent to the contact planes. The classification problem in Legendrian knot theory is lot finer than its topological counter part. The problem gets even trickier when we start considering links. In this talk, I'll survey some of the recent results in this area and then discuss the classification problem for cable links. Part of this is joint work with John Etnyre and Tom Rodewald.

15:45 Merlin Christ (Univ. Paris Cité)
From Lefschetz fibrations to sheaves of categories.
Abstract: Suppose we are given an exact symplectic manifold with a Lefschetz fibration to the disc. As shown by Seidel, the Fukaya category can be recovered from the Fukaya-Seidel category of the fibration, built from the Fukaya category of the regular fiber and the Lagrangian vanishing cycles. We will see in this talk how this construction can be reformulated in terms of a perverse sheaf of categories on the disc. One of the advantages of the sheaf language is that it can be applied to Lefschetz fibrations over any surface with boundary. We then specialize to examples of Fukaya categories of Lefschetz fibrations over surfaces due to Ivan Smith, which can be studied quite effectively using this sheaf language. In these examples, standard construction in symplectic geometry, for instance of Lagrangian matching spheres, acquire representation theoretic meanings.

Next symplectix:

6/12 (Haney, Opshtein, Vertesi (postponed)) 10/01 (?, ?, ?)

Other symplectic activity in Paris (and in France):

- CAST 2025 Workshop (6-8 February in Grenoble)
- Séminaire Nantes-Orsay
- Symplectic Zoominar (every Fridays at 15:15 (except Symplectix' Fridays) Paris time)

Symplectix 4 octobre 2024

Location: IHP, room 201

The talks are broadcasted via Zoom:
https://zoom.us/j/89445897442?pwd=QmVZWHBiM2Nwb09yUVJERnYrSHJrUT09

 

10:45 Julien Dardennes (Toulouse)  
The coarse distance from dynamically convex to convex.
Abstract: Chaidez and Edtmair have recently found the first examples of dynamically convex domains in R^4 that are not symplectomorphic to convex domains, answering a long-standing open question. In this talk, we present new examples of such domains without referring to Chaidez-Edtmair's criterion. We also show that these domains are arbitrarily far from the set of symplectically convex domains in ℝ4 with respect to the coarse symplectic Banach-Mazur distance by using an explicit numerical criterion for symplectic non-convexity (joint work with J. Gutt, V. Ramos and J. Zhang). .

14:00 Nick Wilkins (Bonn)  
Quantum Steenrod powers and Hamiltonian maps.
Abstract: Quantum Steenrod powers are a relatively new tool in the area of symplectic geometry, with surprisingly wide-reaching connections across mathematics. In this talk, we will highlight various applications of quantum Steenrod powers to dynamical systems and C^0 symplectic topology that will appear in upcoming work, joint with E. Shelukhin. In particular, we will extend Shelukhin's previous work to demonstrate a link between uniruledness and the quantum deformation of the quantum Steenrod power of the point class. We will also look at extensions of this result to pseudorotations with hyperbolic periodic points. We will provide new criteria for the existence of infinitely many periodic points of Hamiltonian diffeomorphisms, using properties of the quantum Steenrod power. Finally, we will demonstrate lower bounds for the Hofer and C^0-norms of iterations of Hamiltonian diffeomorphisms, similarly using properties of the quantum Steenrod power.

15:45 Dylan Cant (Orsay)
Eternal classes in symplectic cohomology.
Abstract: I will present work in progress on certain special classes in symplectic cohomology. The classes under consideration lie in the image of every continuation map (for this reason, we call them eternal classes as they are never born). We give criteria for existence and non-existence of eternal classes. Non-eternal classes in symplectic cohomology can be used to define spectral invariants for contact isotopies of the ideal boundary. The spectral invariants of non-eternal classes behave sub-additively with respect to the pair-of-pants product. This is used to define a spectral pseudo-metric on the universal cover of the group of contactomorphisms..

Next symplectix:

8/11 (Atallah, Chatterjee, Christ), 6/12 (Haney, Opshtein, Vertesi)

Other symplectic activity in Paris:

- Séminaire Nantes-Orsay
- Symplectic Zoominar (every Fridays at 15:15 (except Symplectix' Fridays) Paris time)