### Symplectix 3 décembre 2021

Lieu: IHP, salle 201 le matin et amphi Darboux l'après-midi

The seminar will take place in presence, but will be broadcasted via zoom

10:45 Anna Florio (Dauphine)
Right-handed flows on $S^3$ and a quantitative condition to find explicit examples.

Abstract: Global surfaces of section for Reeb flows in dimension 3 are powerful tools to study the associated dynamics. Among Reeb flows on the 3-sphere, those that are right-handed in the sense of Ghys exhibit lots of such global surfaces of section: indeed, Ghys proved that for a right-handed flow any finite (non-empty) collection of periodic orbits is the boundary of a global surface of section. In a joint work with U. L. Hryniewicz, we give a sufficient condition to assure that a flow is right-handed. After presenting our condition, we discuss applications, notably to strictly convex energy levels of Hamiltonian flows on $\R^4$ and to Riemannian geodesic flows on the 2-sphere.

14:00 Cheuk-Yu Mak (Edimbourgh)
Quasimorphisms and Lagrangian links on the 2-sphere
Abstract: When a group G is simple, there is no non-trivial homomorphism from G to the real line but G can possibly have a lot of quasimorphisms. It makes quasimorphisms an important concept in studying groups. In particular, it is an interesting question to ask how many quasimorphisms a group has. In this talk, I will explain how to use Floer theory to construct uncountably many linearly independent homogeneous quasimorphisms on the area-preserving diffeomorphism/homeomorphism group of the 2-sphere, which answers a question of Entov-Polterovich-Py. This is based on a joint work with  Daniel Cristofaro-Gardiner, Vincent Humilière, Sobhan Seyfaddini and Ivan Smith.

15:45 HDR defense of Maxime Zavidovique (Jussieu)
Discrete Weak KAM Theory: an introduction through examples and its applications to twist maps.

Prochaines séances: 7 janvier (A. Keating, E. Cineli, ?), 4 février (M. Bertelson, ?, ?)

Autre activité symplectique à Paris:

- Séminaire Nantes-Orsay (à Nantes le 10 décembre)

### Symplectix 12 novembre 2021

Lieu: IHP, amphi Hermite.

The seminar will take place in presence, but will be broadcasted via zoom

10:45 Bingyu Zhang (Grenoble)
On computing of the Chiu-Tamarkin complex.
Abstract: The Chiu-Tamarkin complex is a cohomology theory of open sets in cotangent bundles, which is defined using microlocal sheaves. I will explain its computation on convex toric domains. The value for a sequence of capacities defined using the Chiu-Tamarkin complex is then demonstrated using the computation. If time permits, I will go over recent computation on the unit disk bundle.

14:00 Vincent Colin (Nantes)
On the genericity of Birkhoff sections and topological entropy for Reeb vector fields in dimension 3.
Abstract: In a joint work with Pierre Dehornoy and Ana Rechtman we proved that in dimension 3 every nondegenerate Reeb vector field is carried by a broken book decomposition. Together with Dehornoy, Hryniewicz and Rechtman, we elaborate on this result to prove that C^1 or C^\infty generically, depending on cases, Reeb vector fields admit Birkhoff sections. We also obtain the genericity of topological entropy.

15:45 Benoit Joly (Bochum)
Barcodes for Hamiltonian homeomorphisms of surfaces
Abstract: In this talk, we will study the Floer Homology barcodes from a dynamical point of view. Our motivation comes from recent results in symplectic topology using barcodes to obtain dynamical results. We will give the ideas of new constructions of barcodes for Hamiltonian homeomorphisms of surfaces using Le Calvez's transverse foliation theory. The strategy consists in copying the construction of the Floer and Morse Homologies using dynamical tools like Le Calvez's foliations.

Prochaines séances: 3 décembre (A. Florio, C.-Y. Mak, HDR Zavidovique), 7 janvier (A. Keating, E. Cineli, ?), 4 février (M. Bertelson, ?, ?)

Autre activité symplectique à Paris:

### Symplectix 1er octobre 2021

Lieu: IHP, amphi Hermite.

The seminar will take place in presence, but will be broadcasted via the zoom link:

10:45 Klaus Niederkrüger (Lyon)
An overtwisted convex hypersurface in dim>3.
Abstract:. The generalization of the notion of "overtwisted contact structures" to dim>3 by Borman-Eliashberg-Murphy (BEM) has been of huge impact for high dimensional contact topology.  Their definition is extremely technical though, and it is only due to Casals-Murphy-Presas that we dispose of necessary tools to recognize some real-life examples of overtwisted (in the sense of BEM) contact manifolds.
Nonetheless, we feel that among the equivalent definitions found so far, there is none that is as basic as the 3-dimensional overtwisted disk.
In my talk, I will explain why a certain convex hypersurface is overtwisted, and what would still be needed to be understood to claim that this hypersurface is "the overtwisted disk" in higher dimension

14:00 Thibaut Mazuir (Jussieu)
Higher algebra of A-infinity algebras in Morse theory
Abstract: In this talk, I will introduce the notion of n-morphisms between two A-infinity algebras. These higher morphisms are such that 0-morphisms correspond to standard A-infinity morphisms and 1-morphisms correspond to A-infinity homotopies. The set of higher morphisms between two A-infinity-algebras then defines a simplicial set which has the property of being a Kan complex. The combinatorics of n-morphisms are moreover encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra.
Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of A-infinity algebras naturally arises in the context of Morse theory.

15:45 Claude Viterbo (Orsay) - To be confirmed
Stochastic Homogenization for Hamilton-Jacobi equations
Abstract: To a Hamiltonian microscopically periodic on  $\mathbb R^{2n}$, we can associate a macroscopic Hamitlonian depending only on the variable $p$. This result is due to Lions-Papanicolaou-Varadhan for viscosity solutions and to the author for variational solutions. We shall deal here for variational solutions with the case of a random Hamiltonian and the conclusion is similar : we see macroscopically an effective Hamiltonian $\overline H (p)$.  In the viscosity case, this is only known for $H$ convex in $p$ (Rezakhanlou-Tarver and Souganidis). The proof involves besides symplectic topological methods, some approximation methods for non-smooth Hamiltonians and some old results on the structure of compact abelian groups.

Prochaines séances: 12 novembre, 3 décembre.

Autre activité symplectique à Paris:

- Exposé de Vivek Shende au séminaire d'analyse algébrique de Jussieu le 18 octobre à 14h.
-
Séminaire Nantes-Orsay