Symplectix 8 janvier 2021



10h45 : Simon Allais (ÉNS Lyon)

Title: Periodic points of Hamiltonian diffeomorphisms and generating functions

Abstract: Ginzburg and Gürel recently showed that a hamiltonian diffeomorphism of CP^d a hyperbolic periodic point have infinitely many periodic points whereas fixed points of a pseudo-rotation are isolated as an invariant set. In 2019, Shelukhin proved a homology version of the Hofer-Zehnder conjecture in a large class of symplectic manifolds M that includes CP^d: a Hamiltonian diffeomorphism with more homologically visible fixed points than the dimension of the homology of M has infinitely many periodic points. These results rely on the quantum structure of the Floer homology.
In this talk, I will explain how the study of sublevel sets of generating functions can replace the use of J-holomorphic curves and Floer theory in the study of periodic points of CP^d, based on ideas of Givental and Théret in the 90s.

13h45 : Alexandre Jannaud (Neuchâtel)

Title : Dehn-Seidel twist, C^0 symplectic geometry and barcodes

Abstract : In this talk I will present my work initiating the study of the $C^0$ symplectic mapping class group, i.e. the group of isotopy classes of symplectic homeomorphisms, and present the proofs of the first results regarding the topology of the group of symplectic homeomorphisms. For that purpose, we will introduce a method coming from Floer theory and barcodes theory.

Applying this strategy to the Dehn-Seidel twist, a symplectomorphism of particular interest when studying the symplectic mapping class group, we will generalize to $C^0$ settings a result of Seidel concerning the non-triviality of the mapping class of this symplectomorphism. We will indeed prove that the generalized Dehn twist is not in the connected component of the identity in the group of symplectic homeomorphisms. Doing so, we prove the non-triviality of the $C^0$ symplectic mapping class group of some Liouville domains.