Symplectix 5 avril 2024

Location: IHP, room 201

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

 
10:45 David Bechara Senior (Aachen) 
The asymptotic action of area preserving disk maps and some of its properties.
Abstract: Given a diffeomorphism of the disk that preserves the standard symplectic form, I will introduce the asymptotic action associated to this map. I will then show a pointwise formula relating the asymptotic action to the asymptotic winding number of pairs of points. As a corollary one obtains a generalization for a well known result by A. Fathi which gives a formula for the Calabi invariant of a disk map in terms of its mean winding numbers. Additionally I will focus on how this formula can be used to study symplectic dynamical information for irrational pseudo-rotations of the disk. This talk includes joint work with Patrice Le Calvez and Abror Pirnapasov.
 

14:00 Yanki Lekili (London)
New orders associated with terminal threefold singularities.
Abstract: This is based on joint work with Jenia Tevelev. I will tell you a story of a vector bundle due to Kawamata that captures the "categorical essence" of certain terminal 3-fold singularities. Our computations are done in the mirror via Floer theory but it teaches us (with a proof!) something new about derived categories of terminal threefold singularities and noncommutative algebra by providing explicit deformations of the so-called Kalck-Karmazyn algebra associated with cyclic quotient surface singularities.

15:45 Salammbo Connolly (Orsay)
On torsion in (bi)linearized Legendrian contact homology.
Abstract: Given a Legendrian submanifold in a contact manifold, one can define its Legendrian Contact Homology (LCH). This invariant is unfortunately difficult to compute. However, there exist two more computable variants, linearized LCH (defined by Chekanov), and bi-linearized LCH (defined by Bourgeois and Chantraine). For legendrian knots in R^3, very little is known about the possibility of having torsion in these invariants when they are defined over integer coefficients. In joint work with Frédéric Bourgeois, we give properties of torsion that can appear in linearized LCH with integer coefficients, and also give the full geography of bi-linearized LCH with integer coefficients.

Next Symplectix:
 

26/04 (Coté, Ward, Dimitroglou-Rizell)

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)

Mini-cours Shaoyun Bai

Lectures on recent developments in Hamiltonian Floer homology

     by Shaoyun Bai (Columbia university)

Abstract: This lecture series will provide an introduction to recent conceptual and technical advances in Hamiltonian Floer theory.

 

Lecture one (Lundi 11 de 10h00 à 11h30, à Jussieu 15-16-413): In the first lecture, we will discuss how to organize moduli spaces from Floer theory into structures like flow categories (gradient flow lines), flow bimodules (continuation maps), and flow multi-modules (algebraic structures). Then, we will indicate how coherent Kuranishi-type structures can be defined and why they are relevant to defining virtual counts.

Lecture two (Mercredi 13 de 14h30 à 16h00, à jussieu 15-25-502): In the second lecture, we will focus on explaining the construction of global Kuranishi charts of various types of moduli spaces arising in Hamiltonian Floer theory.

Lecture three (jeudi 14 de 15h30 à 17h, à Orsay, salle 2L8): In the final lecture, we will introduce a perturbation scheme, known as the Fukaya-Ono-Parker perturbations, which allows one to define Hamiltonian Floer theory with integer coefficients. The principle will be sketched, emphasizing how to use this theory as a black box.

 

Symplectix 1er mars 2024

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)