Symplectic topology, contact topology and interactions, Paris

10h45 : Sushmita Venugopalan (IMSc Chennai) : Tropical Fukaya Algebras

Abstract: A multiple cut operation on a symplectic manifold produces a collection of cut spaces, each containing relative normal crossing divisors. We explore what happens to curve count-based invariants when a collection of cuts is applied to a symplectic manifold. The invariant we consider is the Fukaya algebra of a Lagrangian submanifold that is contained in the complement of relative divisors. The ordinary Fukaya algebra in the unbroken manifold is homotopy equivalent to a `broken Fukaya algebra' whose structure maps count `broken disks' associated with rigid tropical graphs. Via a further  degeneration, the broken Fukaya algebra is homotopy equivalent to a `tropical Fukaya algebra' whose structure maps are sums of products over vertices of tropical graphs. This is joint work with Chris Woodward.

13h45 : Steven Sivek (Imperial College London) : TBA

10h45 : Pierre Dehornoy (U. Grenoble Alpes) : The genus-one question for open book decomposition

Abstract: In dimension 3, every contact structure is supported by an open book decomposition. When the structure is overtwisted one can strengthen the statement and have an OBD with genus-zero pages. For tight structures, there are many examples of OBD with genus-one pages, and Etnyre asked around 2006 whether it always exists. The question is still open. I will discuss a parallel and related question, for Anosov flows instead of contact structures. I will explain some constructions of genus-one OBD, and report on a partial result for OBD with genus-two pages (with the assistance of a computer).

13h45 : Octav Cornea (U. de Montréal) : Triangulation and persistence

10h45 : Alfonso Sorrentino (U. Rome Tor Vergata) : Rigidity phenomena in Billiard and Hamiltonian Dynamics

Abstract: In this talk I shall address several rigidity questions appearing in the study of  billiard maps and Hamiltonian dynamical systems, with a particular focus on their integrability and their action-spectral properties.

13h45 : Yusuke Kawamoto (ENS & Sorbonne Université) : Around a Question of Entov-Polterovich-Py

10h45 : Alberto Abbondandolo (Bochum) : Bi-invariant Lorentz-Finsler structures on the linear symplectic group and contactomorphism group

Abstract: It is well known that the linear symplectic group and the contactomorphism group do not admit any bi-invariant metric which is compatible with the Lie group topology. In this talk, I will discuss two mutually related bi-invariant Lorentz-Finsler structures on these groups. The talk is based on some work in progress with Gabriele Benedetti and Leonid Polterovich.

13h45 : Marco Mazzucchelli (ENS Lyon) : A few properties of Besse contact manifolds.

Abstract : A closed connected contact manifold is called Besse when all of its Reeb orbits are closed, and Zoll when furthermore all Reeb orbits have the same minimal period. In this talk, I will present a recollection of recent results/work in progress on the subject:
- It is known that Besse contact 3-spheres are strictly contactomorphic to rational ellipsoids. In higher dimensions, the analogous statement is open. Nevertheless, I will show that at least those contact (2n-1)-spheres that are convex hypersurfaces in symplectic vector spaces still "resemble" a rational ellipsoid. This is joint work with Marco Radeschi.
- Inspired by recent results on the systolic optimality of Zoll contact manifolds, I will show that Besse contact 3-manifolds are local maximizers of a suitable generalized systolic ratio. This is joint work with Alberto Abbondandolo and Christian Lange.

10h45 : Baptiste Chantraine (Nantes) : compact objects in the Fukaya category and representations of Eliashberg-Chekanov algebra.

Abstract : Let L be a compact Lagrangian in a Weinstein manifold obtained from a subcritical one by attaching a handle along a Legendrian V. We will see how to associate to L a filling of a satelite of V and how this one induces a representation of the Chekanov-Eliashberg algebra of V. We will show that Legendrian contact homology linearised with respect to this representation recovers the Floer homology of L. We will talk about extensions of this considerations to A-infinity opérations on both sides. This is a joint work with G. Dimitroglou-Rizell and P. Ghiggini
 

13h45 : Paolo Ghiggini (Nantes) : Many real projective spaces are not Liouville fillable

Abstract : I will show that the standard contact structure on the real projective spaces RP^{4k+1} is not Liouville fillable using a classical argument on degeneration of moduli spaces of holomorphic spheres. A stronger result has been obtained by Zhengyi Zhou using more algebraic methods. This is a joint work with Klaus Niederküger