Symplectix 1er mars

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.

Abstract:
Abouzaid--Groman--Varolgunes constructed a chain-level framed
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)

Symplectix 2 février 2024

Location: IHP, room 201

The talks are broadcasted via Zoom:
https://zoom.us/j/89445897442?pwd=QmVZWHBiM2Nwb09yUVJERnYrSHJrUT09

 
10:45 Patrice Le Calvez (Sorbonne Université)
Periodic orbits of area preserving surface homeomorphism.

Abstract: In a joint work with Pierre-Antoine Guihéneuf and Alejandro Passeggi we will explain why an area preserving homeomorphism of a closed surface isotopic to the identity that has a rotation vector with a rational direction has infinitely many periodic orbits (result independently proved by Rohil Prasad). More precisely, we will explain how associate to every non trivial ergodic measure an ``interval'' of periodic orbits.


14:00
Mihai Damian (Université de Strasbourg)
Morse type theories with differential graded coefficients and symplectic applications.

Abstract:
Motivated by questions on the existence of contractible periodic
characteristics on some hypersurfaces of the cotangent bundle, we
develop a general Morse theory with coefficients in a differential
graded module over the ring of singular chains of the based loop
space. This generalizes previous works of JF. Barraud and O. Cornea.
Joint with JF. Barraud, V. Humiliere and A. Oancea.


15:45 Yuhan Sun (Imperial College)
Cech complex of the symplectic cohomology.
Abstract: Symplectic cohomology behaves like a presheaf under the Viterbo restriction map. For a Liouville domain, if it is covered by Poisson-commuting Liouville subdomains then we will show the symplectic cohomology forms a sheaf on this cover. This enables us to do local-global computations via its Cech complex. Explicit examples will be discussed in the setting of Lagrangian torus fibrations, and of the Cieliebak-Oancea exact sequence for cobordisms. Joint with U. Varolgunes.

Next Symplectix:

01/03 (Berger, Deshmukh, Leclercq), 05/04 (Bechara, Lekili), 26/04 (Coté, Ward, ?)


Other symplectic activity in Paris:

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

Symplectix 12 Janvier 2024

Location: IHP, amphi Hermitte

The talks are broadcasted via Zoom:
https://zoom.us/j/89445897442?pwd=QmVZWHBiM2Nwb09yUVJERnYrSHJrUT09

 

10:45 Pranav Chakravarthy (Bruxelles)
Homotopy type of Equivariant symplectomorphism groups and Embedding spaces.

Abstract: In this talk, we present results on the homotopy type of the group of equivariant symplectomorphisms of S²xS² and CP² blown up once under the presence of a Hamiltonian circle and finite cyclic group actions. We prove that the group of equivariant symplectomorphisms is homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the group action extends to a single toric action or to exactly two nonequivalent toric actions.
Time permitting, we shall also discuss upcoming work on homotopy type of equivariant embedding spaces and their relation to symplectomorphism groups. This talk is based on joint work with Martin Pinsonnault and upcoming work with River Chiang, Liat Kessler and Martin Pinsonnault.


14:00
Fabio Gironella (Nantes)
Vanishing cycles for symplectic foliations.

Abstract:
The main objects of the talk will be symplectic foliations, and more precisely a subclass of these called "strong". Strong symplectic foliations are meant to be one of the possible rigid generalizations of taut foliations to high dimensions, and indeed have quite a rigid nature, with techniques such as pseudo-holomorphic curves à la Gromov and asymptotically holomorphic sequences of sections à la Donaldson working well in this setting. I will present a joint work (in progress) with Klaus Niederkrüger and Lauran Toussaint that aims at giving a new obstruction for a symplectic foliation to be strong. This comes in the form of a symplectic high-dimensional version of vanishing cycles for smooth codimension 1 foliations on 3-manifolds, and the proof relies on pseudo-holomorphic curve techniques, in a way which is parallel to the case of Plastikstufe introduced by Niederkrüger '06 in the contact case.


15:45 Bingyu Zhang (Odense)
Hochschild cohomology of Tamarkin category and symplectic (co)homology.
Abstract: We can define a Tamarkin category for an open set in a cotangent bundle using the microlocal geometry of sheaves. Some computational evidence indicates that the so-called Chiu-Tamarkin invariant is deeply related to the Hochschild cohomology of the Tamarkin category and symplectic (co)homology. In this talk, we will explain the relation precisely: All of them are isomorphic in a suitable filtered sense. The talk is based on a joint work with Christopher Kuo and Vivek Shende..

Next Symplectix:

02/02 (M. Damian, Y. Sun, ?), 01/03 (?, ?, ?), 05/04 (?, ?, ?) 


Other symplectic activity in Paris:

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