Working seminar on symplectic topology and related topics

• Nov 9: Vincent Humilière
    Title: Relation between periodic Floer homology and link Floer homology (after Guanheng Chen) I
    Abstract: Link Floer homology and Periodic Floer homology are both invariants of area preserving diffeomorphisms of surfaces that were involved in recent progress on C-infinity closing lemma and simplicity conjecture. The goal of the talk is to learn about the papers arxiv:2111.11891 and arxiv:2209.11071 by G. Chen which establishes a relation between these two theories. In the first part we will review link Floer homology, which is a Lagrangian Floer theory in d-fold symmetric products (dimension = 2d) and see that it admits an alternative definition that uses curves in a 4 dimensional manifold. This is due to Chen and inspired by Lipshitz's interpretation of Heegaard Floer theory.
• Nov 16: Vincent Humilière
    Title: Relation between periodic Floer homology and link Floer homology (after Guanheng Chen) II
• Nov 23: Francesco Morabito
    Title: Hofer Pseudonorms on Braid Groups and Quantitative Heegaard-Floer Homology
    Abstract: Given a lagrangian link with k components it is possible to define an associated Hofer pseudonorm on the braid group with k strands. In this talk we are going to detail this definition, and explain how it is possible to prove non degeneracy if k=2 and certain area conditions on the lagrangian link are met. The proof is based on the construction, using Quantitative Heegaard-Floer Homology, of a family of quasimorphisms which detect linking numbers of braids on the disk.
• Nov 30: Matija Sreckovic
    Title: Higher-Dimensional Heegaard-Floer Homology
    Abstract: The goal of my talk will be to give a survey of the paper "Applications of higher-dimensional Heegaard-Floer homology to contact topology" by V. Colin, K. Honda and Y. Tian (arxiv:2006.05701 ). In the first part of the talk, I will define the higher-dimensional Heegaard-Floer homology groups (HDHF) associated to a Weinstein domain W and a symplectomorphism h of W which restricts to the identity on the boundary.
  In the second part of the talk, I will define the contact class in HDHF and explain how it can be applied to detect non-Liouville-fillability of some contact manifolds, as well as the existence of closed Reeb orbits. If time permits, I will also say a few words about the variant of symplectic Khovanov homology defined in this paper.
• Dec 7: No seminar
• Dec 14: No seminar (the speaker caught covid...)
• Jan 18: Dustin Connery-Grigg
    Title: Understanding the geometry of Hamiltonian Floer complexes of Hamiltonian isotopies on surfaces (Part 1)
    Abstract: In arXiv:2102.11231 I explained how, for Hamiltonian isotopies on surfaces, one can use ideas originating in Hofer-Wysocki-Zehnder’s work on finite energy foliations (along with later developments due to Siefring) together with the topology of capped braids in order to gain significant insight into the topological behaviour of various collections of Floer-type cylinders which are relevant to fundamental constructions in Floer theory. Some notable applications of this theory are the provision of an explicit bridge between Floer theory and Le Calvez’s theory of transverse foliations, as well as both motivating the introduction of a novel class of spectral invariants in addition to providing a purely topological characterization of the most important of these. In these two talks, I will aim to explain the main ideas and details of this theory.
• Jan 25: Dustin Connery-Grigg
    Title: Understanding the geometry of Hamiltonian Floer complexes of Hamiltonian isotopies on surfaces (Part 2)
    Abstract: Continuation of Part 1
• Feb 1: Vukasin Stojisavljevic
    Title: An introduction to topological entropy
    Abstract: This will be an introductory talk on topological entropy. We will start from the definition, discuss basic properties of topological entropy and illustrate the relevant notions in certain classical examples. We will also briefly discuss theorems of Yomdin and Newhouse which relate topological entropy of smooth maps on compact manifolds to the volume growth of subsets. This will be the first in a series of talks on topological entropy, which aim at understanding the recent paper of Ginzburg, Gurel and Mazzucchelli - https://arxiv.org/abs/2212.00943.
• Feb 8: Erman Cineli
    Title: On the barcode entropy of geodesic flows
    Abstract: Recently, in https://arxiv.org/pdf/2212.00943.pdf, Ginzburg, Gurel and Mazzucchelli introduced and studied the barcode entropy for geodesic flows of closed Riemannian manifolds. They show that the barcode entropy bounds from below the topological entropy of the geodesic flow, and for Riemannian metrics on surfaces, it is equal to the topological entropy. The goal of this talk is to go through the main steps of their proof of the former result (the inequality) above.
• Feb 15: Erman Cineli and Vukasin Stojisavljevic
    Title: On the barcode entropy of geodesic flows II
    Abstract: The first half of the talk will be a continuation of the previous lecture by Erman Çineli. In the second half of the talk, we will discuss a result from the same paper - https://arxiv.org/pdf/2212.00943.pdf, which states that barcode entropy equals topological entropy for geodesic flows on surfaces.
• Feb 22: No seminar
• Mar 1: Marcelo Alves
    Title: An introduction to categorical entropy and some applications to symplectic dynamics
    Abstract: I will give an introduction to the categorical entropy of endofunctors of a triangulated category, introduced by Dimitrov, Haiden, Katzarkov and Kontsevich. My aim will be to explain a connection between the categorical entropy and the topological entropy of certain symplectomorphisms which appears in the recent work of Bae and Lee.
• Mar 8: Ibrahim Trifa
    Title: Lagrangian Quantum Homology and the Superpotential
    Abstract: A standard method of proving that a Lagrangian submanifold has non-zero quantum (or Floer) homology is to find critical points of a "superpotential". The purpose of this talk will be to understand Biran and Cornea's construction of this superpotential in the context of Lagrangian Quantum Homology (defined with a pearl complex), and how it can detect non-zero homology. If time permits, I will also present the superpotential's original construction by Fukaya, Oh, Ohta and Ono.
     Ibrahim's notes
• Mar 15 (WARNING: we will start exceptionally at 15:00): Yusuke Kawamoto
    Title: Isolated hypersurface singularities, spectral invariants, and quantum cohomology
    Abstract: We discuss the relation between hypersurface singularities (e.g. ADE, E₆,E₇,E₈, etc) and spectral invariants, which are symplectic invariants coming from Floer theory.
• Mar 22: Baptiste Serraille
    Title:The sharp C⁰-fragmentation property for Hamiltonian diffeomorphisms and homeomorphisms on surfaces.
    Abstract: Fragmentation properties have been used by Banyaga, Fathi and Thurston in order to study the algebraic structures of groups of diffeomorphisms, volume preserving diffeomorphisms/homeomorphisms and Hamtilonian diffeomorphisms. This property has been improved on surfaces by Entov, Polterovich and Py and later by Seyfaddini to C⁰-fragmentation in order to construct (Calabi-)quasi-morphisms. I will present how it is possible to have a sharper version of those results and hopefully dive into the cute technical details that arise in the proof.
• Mar 29: Agustin Moreno
    Title: On the three-body problem.
    Abstract:I will give an introduction to the classical three-body problem, but from the modern perspective of symplectic and contact geometry.
• April 5: Dusan Joksimovic
    Title: Rigidity of coisotropic submanifolds
    Abstract: Let $N$ be a closed coisotropic submanifold of a given symplectic manifold $(M,\omega).$ In the first part of the talk, I will introduce a construction of a Lagrangian embedding of $N$ into a certain symplectic submanifold of $(M\times N, \omega \oplus -\omega \vert_N).$ The second part of the talk is devoted to applications. I will show how one can use the above construction to provide solutions to the problems of the existence of leafwise fixed points and rigidity of coisotropic embeddings.
• Apr 19: Francesco Morabito
    Title: Equivariant pairs of paints
    Abstract: In this talk we are going to sketch the proof of a symplectic Smith-type inequality, following a construction detailed by Paul Seidel in his paper from 2015 "The Equivariant Pair-of-Pants Product in Fixed Point Floer Cohomology". In order to prove the theorem, we are going to define Tate cohomology of a group with values in a chain complex, an equivariant version of Floer cohomology and a binary product which counts equivariant pairs of pants between Hamiltonian closed orbits.