Groupe de travail de géométrie symplectique et de contact

Co-organisé par Ella Blair et Simon Vialaret.
Tous les jeudis à 15h30, en salle 2L8.

Séances passées

• 11/01/2024 : An overview of Hamiltonian Floer homology (Julio Sampietro Christ)
  
  Abstract : This talk is meant to be a broad introduction to Hamiltonian Floer homology. We will review the main ideas, as well as the challenges (transversality, compactness, etc.), that come into play for defining Floer homology groups.
  
  
• 18/01/2024 : A Floer Fundamental Group after Jean-François Barraud (Salammbo Connolly)
  
  Abstract : In his paper "A Floer Fundemental Group", Jean-François Barraud exhibited a desciption of the fundemental group of a symplectic manifold using Floer theory, using a similar Morse-theoretic description as a guiding force. The goal of this talk will be to give an overview of both the Morse and Floer constructions to shed light on how both theories provide more information than just the that of their respective homologies.
  
  
• 25/01/2024 : Pas de séance



• 01/02/2024 : Introduction to open books decompositions (Simon Vialaret)
  
  Abstract : To prepare for Ella's talk, i will introduce open book decomposition for contact manifolds, give examples and state Giroux correspondence.



• 08/02/2024 : Pas de séance



• 15/02/2024 : Donaldson techniques for the existence of open book decompositions (Ella Blair)
  
  Abstract : We will discuss a classic technique: Donaldson’s asymptotically holomorphic sections. The aim of this discussion will be to understand the general idea of this technique and how it was used to prove the existence of adapted open book decompositions on closed contact manifolds.



• 22/02/2024 : Pas de séance
• 29/02/2024 : Introduction to symplectic homology (Baptiste Seraille)
  
  Abstract : We will define the symplectic homology $SH(W,\lambda)$ of a Liouville manifold $(W, \lambda)$. We will try to explain how this definition is trying to count Reeb orbits of the contact manifold $(\partial W, \lambda)$. The well-definition of the symplectic homology relies on tools of Floer homology that we already saw and a specific tool (also useful in other situations) a maximum principle. I will try to sketch some proofs if time permit.



• 07/03/2024 : Proof of Viterbo's isomorphism (Jianqiao Shang )
  
  Abstract : We will study how to construct homomorphisms between two Morse homologies and some interesting examples. Then apply the idea to prove the Viterbo's isomorphism: $SH(DT^*M)$ is isomorphic to $H(\Omega M)$, the singular homology of the loop space. As a preliminary, I will also briefly introduce the Morse theory of loop space on Riemannian manifolds.



• 21/03/2024 : Link spectral invariants (Ibrahim Trifa)
  
  Abstract : Link spectral invariants were introduced by Cristofaro-Gardiner, Humilière, Mak, Seyfaddini and Smith in 2022 to solve the simplicity conjecture for any oriented closed surface: they showed that the Hamiltonian homeomorphism group of a closed surface is never simple. In my first talk, I will explain the construction of those invariants. The second talk will be focused on some applications.



• 28/03/2024 : Link spectral invariants part 2 (Ibrahim Trifa)



• 04/04/2024 : The Flow trees and Legendrian submanifolds (Filip Strakos)
  
  Abstract : We will care about Legendrian submanifolds in one-jet spaces (or exact Lagrangian immersions with transverse intersections in cotangent bundles). Legendrian isotopy classes of Legendrian submanifolds in a (nice) contact manifold can be studied using Legendrian contact homology (LCH), which is a Legendrian isotopy invariant that belongs to the SFT framework (so we are solving the CR equation without Hamiltonian perturbation, due to Chekanov (for dimension 1) and Ekholm, Etnyre, Sullivan (for any dimension)). The LCH is the homology of the so-called Chekanov-Eliashberg algebra, where the differential counts rigid pseudo-holomorphic disks with boundary punctures mapping boundary on the Legendrian submanifolds and with asymptotic approaching to Reeb chords (or intersection points of the immersed Lagrangian). It is a beautiful observation of Ekholm that the contribution to the differential can be obtained purely from Morse theory (of something that resembles local multisection of the one-jet bundle). Therefore, the problem of solving the CR-equation is reduced to combinatorics. We will define the flow tree framework and compute some interesting examples (Hopf-link of arbitrary dimension by Bourgeois-Galant, Lambert-Cole product of unknots) providing a necessary amount of technical details. The talks should ideally lead us to the Morse-theoretical interpretation of Adams' Kozsul duality result for simply connected closed manifolds due to Ekholm and Lekili. Beware, there will be a lot of pictures along the way.



• 11/04/2024 : The Flow trees and Legendrian submanifolds part 2 (Filip Strakos)



• 18/04/2024 : Symplectic capacities and two conjectures of Mahler and Viterbo (Simon Vialaret)
  
  Abstract : Symplectic capacities are quantitative symplectic invariants giving obstructions on the existence of symplectic embeddings between two given domains. A well-studied conjecture of Viterbo gives an upper bound on the Hofer-Zehnder capacity of a convex domain in term of its symplectic volume. After introducing symplectic capacities and the conjecture of Viterbo, I will review a result of Arstein-Avidan, Karasev and Ostrover stating that Viterbo conjecture implies another conjecture in convex geometry, Mahler conjecture, seemingly not related to symplectic geometry. Mahler conjecture gives a sharp lower bound on the volume product of a convex domain, and gives a full description of the minimizers. The proof that Viterbo implies Mahler involves a billiard interpretation of the Reeb flow on the boundary of some Lagrangian products, allowing to compute the Hofer-Zehnder capacity of those domains.



• 23/05/2024 : A G-tour of equivariant Floer homology (Julio Sampietro-Christ)
  
  Abstract : The point of these two talks is to give an exposition on Guillem Cazassus' equivariant Lagrangian Floer theory. We will start with an overview of equivariant (co)homology and its Morse version and a review of (cascade) Lagrangian Floer homology. Motivated by the pushforward in Morse theory, we will explore the theory of pseudoholomorphic quilts following Wehrheim and Woodward's papers. Finally, we will construct the equivariant Floer homology groups of two G-Lagrangians and state their basic properties. If time permits, we will discuss some applications and new results in the theory.



• 30/05/2024 : Applications of contact geometry to the 3-body problem (Jianqiao Shang)
  
  Abstract : In this talk, we will see how to construct a contact structure on the (regularized planar circular restricted) three-body problem. As a corollary, we can show the existence of periodic orbits with any energy. This gives a way to save the lives of 'Trisolarans'.