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)

Symplectix 28 juin 2024

 Location: IHP, room 201

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

 
9:30 Matija Sreckovic (ENS)  
Link between Flow Categories of Morse Functions and Fukaya
Categories of Lefschetz Fibrations on the Cotangent Bundle.
Abstract: I will start the talk by explaining some concepts used in Emmanuel
Giroux's construction of a Lefschetz fibration on the cotangent bundle
which extends a given Morse function on the zero section. I will then
state a variant of a conjecture by Paul Seidel on a link between the flow
category of the Morse function and the directed Donaldson-Fukaya category
of the Lefschetz fibration. In the main part of the talk, I will explain
how to prove this conjecture in dimension 2, and some progress I've made
in dimension 3. The main tool in the proof is an explicit handle
decomposition of the real regular fibers of the Lefschetz fibration, which
allows us to see what the vanishing cycles look like.

10:45 Simon Vialaret (Bochum and Orsay)  
Systolic inequalities for S^1-invariant contact forms.
Abstract: In Riemannian geometry, a systolic inequality aims to give a uniform bound on the length of the shortest closed geodesic for metrics with fixed volume on a given manifold. This notion generalizes to contact geometry, replacing the geodesic flow by the Reeb flow, and the length by the period. As opposed to the Riemannian case, it is known that there is no systolic inequality for general contact forms on a given contact manifold. In this talk, I will state a systolic inequality for contact forms that are invariant under a circle action in dimension 3, and give applications to Finsler geodesic flows and to a conjecture of Viterbo.

14:00 Adrien Currier (Nantes)
About the nearby Lagrangian conjecture in locally conformally symplectic geometry.
Abstract: Locally conformally symplectic (lcs) geometry is a generalization of symplectic geometry in which a manifold is endowed with a non-degenerate 2-form that is locally a symplectic form up to some positive factor. If the local behavior of such a manifold is largely identical to that of a symplectic manifold, the global behavior can nonetheless vastly differ. For example, while it is possible to define Lagrangian submanifolds in lcs geometry, we also have to contend with the fact that S^3 \times S^1 has a canonical "exact" lcs structure given  by the canonical contact form of S^3 through a process known as circular lcs-ization.
The foremost goal of this talk will be to familiarize the public with lcs geometry and its ties to other branches geometry, most notably contact geometry. To do this, I will use a couple of results I have obtained during my thesis as a narrative thread. These results will focus on the nearby Lagrangian conjecture in lcs geometry and, more specifically, on the possibility of an lcs adaptation of the Abouzaid-Kragh theorem.

Other symplectic activity in Paris:

- Séminaire Nantes-Orsay
- Symplectic Zoominar (every Fridays except Symplectix' Fridays at 15:15, Paris time)
- Soutenance de thèse de Francesco Morabito le 26 juin à 16h, Ecole Polytechnique, amphi Becquerel.