Location: IHP, room 201

The talks are broadcasted via Zoom:

https://zoom.us/j/89445897442?pwd=QmVZWHBiM2Nwb09yUVJERnYrSHJrUT09

**10:45** Pierre Berger (Jussieu)** An AbC principle for pseudo-rotations.**

*Abstract*: We construct analytic surface symplectomorphisms with unstable elliptic fixed points; this solves a problem of Birkhoff (1927). More precisely, we construct analytic symplectomorphisms of the sphere and of the disk which are transitive, with respectively only 2 and 1 periodic points. This solves problems of proposed by Herman (1998), Fayad-Katok (2004) and Fayad-Krikorian (2018).

To establish these results, we introduce a principle that enables to realize, by an analytic symplectomorphism, properties which are C⁰-realizable by the approximation by the conjugacy method of Anosov-Katok.

**14:00** Rémi Leclercq (Orsay)**Essential loops of Hamiltonian homeomorphisms.***Abstract:* In 1987, Gromov and Eliashberg showed that if a sequence of diffeomorphisms preserving a symplectic form C⁰ converges to a diffeomorphism, the limit also preserves the symplectic form, even though this is a C¹ condition. This result gave rise to the notion of symplectic homeomorphisms, i.e. elements of the C⁰-closure of the group of symplectomorphisms in that of homeomorphisms, and started the study of "continuous symplectic geometry".

In this talk, I will present recent progress in understanding the fundamental group of the C⁰-closure of the group of Hamiltonian diffeomorphisms in that of homeomorphisms. More precisely, I will explain a sufficient condition which ensures that certain essential loops of Hamiltonian diffeomorphisms remain essential when seen as "Hamiltonian homeomorphisms". I will illustrate this method (and its limits) on toric manifolds, namely complex projective spaces, rational products of 2-spheres, and rational 1-point blow-ups of CP².

Our condition is based on (explicit) computation of the spectral norm of loops of Hamiltonian diffeomorphisms which is of independent interest. For example, in the case of 1-point blow-ups of CP², I will explain a surprising behavior of the spectral norm which heavily depends on the choice of the symplectic form. This is joint work with Vincent Humilière and Alexandre Jannaud..

**15:45** Yash Deshmukh (Bonn)

Algebraic structures on relative symplectic cohomology.

Algebraic structures on relative symplectic cohomology.

*Abouzaid--Groman--Varolgunes constructed a chain-level framed*

Abstract:

Abstract:

E2 structure on relative symplectic cohomology. In this talk, I will

outline an extension of this structure to operations parameterized by

curves of all genera and with multiple inputs and outputs. Additionally,

I will discuss the extension of this structure to include operations

from nodal curves satisfying the so-called 'plumber's condition'. I will

indicate how, in different contexts, this structure turns out to be

related to Gromov-Witten CohFTs and the graded Frobenius algebra

structure on Rabinowitz Floer theory.

**Next Symplectix:**

05/04 (Bechara, Lekili, Connolly), 26/04 (Coté, Ward, ?)

**Other symplectic activity in Paris:**

- Mini-course of Shaoyun Bai in the week 11/03-14/03

-** **Séminaire Nantes-Orsay

- Symplectic Zoominar (every Fridays except Symplectix' Fridays at 15:15, Paris time)