Séance du 4 novembre 2016

Lieu: IHP, salle 201


11:00  Florent Balacheff (Lille) 
Systolic contact geometry.
Abstract: Systolic geometry involves a lot of ingredients like algebraic topology, metric geometry or conformal techniques for instance. In this talk, after briefly recall part of this background, we will explain why contact geometry is a natural setting for the study of isosystolic inequalities and the new perspectives it offers. This is joined work with J.C. Alvarez Paiva and K. Tzanev.

14:15  Umberto Hryniewicz (Rio de Janeiro) 
Local subharmonic invariants of periodic points of Hamiltonian

Abstract: As is well-known, the variational structure underlying
Hamiltonian systems makes their fixed point theory richer than as
predicted by Lefschetz theory. In this talk I will explore the
variational structure of the discretized action functional to put on
rigorous grounds the existence of new local invariants of periodic
points of Hamiltonian diffeomorphisms. These invariants, which are of
a subharmonic nature, are different from local Floer homology. Their
existence was predicted in previous work as a local contact homology.
Our new point of view allows us to completely understand their
iteration properties by means of an "invariant shifting lemma". As a
byproduct of the methods we establish some transversality results in
Morse homology with finite cyclic group symmetries. This is joint work
with Hein and Macarini.

16:00  Laurent Charles (Paris 6 et ENS) 
Quantum speed limit vs displacement energy.

Abstract: The quantum speed limit is a universal bound on the energy required to pass from one state to another orthogonal state in a quantum system. Similarly, in symplectic topology, the displacement energy is the minimal energy needed to displace a given subset of a symplectic manifold. I will discuss how these two notions are related in the semiclassical limit. Joint work with Leonid Polterovich.

Prochaines séances: 2 décembre (S. Suhr, S. Courte, ?), 6 janvier.

Autre activité symplectique à Paris:
- Séminaire Nantes-Orsay
- Soutenance de thèse Maÿlis Limouzineau: le 21 octobre à 14h à Jussieu.
- Doctorat Honoris Causa de Dusa McDuff : les 10 et 11 octobre à Paris 6.

Séance du 7 octobre 2016

Lieu: IHP, salle 01


11:00  Alvaro Del Pino Gomez (Madrid)
An introduction to Engel structures.
Abstract: A maximally non-integrable 2-plane field in a 4-manifold is called an Engel structure. Like contact structures, they possess a Darboux model, and hence they lack local invariants. At the same time, the lack of a Gray stability type theorem makes the construction of global invariants (up to deformation through Engel structures) a complicated matter.
   The aim of the talk will be to review the recent advances in flexibility in Engel geometry. Namely, I will outline the proof of the existence h-principle for Engel structures proven in [CPPP] and the h-principle for horizontal and transverse immersions proven in [CP]. If time allows, I will discuss some work in progress regarding horizontal knots in standard Engel R^4.
[CP] A. del Pino, F. Presas. Flexibility for tangent and transverse immersions in Engel manifolds. In preparation.
[CPPP] R. Casals, J.L. Pérez, A. del Pino, F. Presas. Existence h-Principle for Engel structures . arXiv:1507.05342.

14:15  Milena Pabiniak (Cologne) 
The contact Arnold Conjecture for lens spaces via a non-linear Maslov index.
Abstract: Diffeomorphisms in symplectic category posses certain rigidity properties. An important manifestation of rigidity is given by the conjectures posed by V. Arnold describing a lower bound for the number of fixed points of a Hamiltonian diffeomorphism of a compact symplectic manifold, greater than what topological arguments could predict.
   Arnold Conjectures present a difficult problem and motivated a lot of important research in symplectic geometry. It has been translated to the contact geometry setting where one looks for a lower bound for the number of translated points.
   Givental's construction of a quasimorphism, called the non-linear Maslov index, allows one to prove the Arnold Conjecture for complex and real projective spaces. Moreover, the properties of this quasimorphism imply that the real projective space is orderable, has a non-displaceable pre-Lagrangian and that its discriminant and oscillation norms are unbounded.
   In this talk I will describe my work joint with G. Granja, Y. Karshon and S. Sandon, aimed at constructing a quasimorphism for lens spaces (building on the ideas of Givental) and proving the corresponding statements for these spaces (Contact Arnold Conjecture, orderability, ... ).   I will discuss the difficulties of constructing such a generalization to lens spaces and the possibility of generalizing these ideas even further: to prequantizations of symplectic toric manifolds.

16:00  Yuichi Ike (Tokyo) 
Categorical localization for the coherent-constructible correspondence 

Abstract: The coherent-constructible correspondence is a version of homological mirror symmetry for toric varieties. It says that the derived category of coherent sheaves on a toric variety is equivalent to the derived category of constructible sheaves on the real torus whose microsupports are contained in some Lagrangian. We prove categorical localization for the constructible-side categories, which can be regarded as a microlocal counterpart of categorical localization for Fukaya categories. This is a joint work with Tatsuki Kuwagaki.

Prochaines séances: 4 novembre (F. Balacheff, U. Hryniewicz, ?), 2 décembre (S. Suhr, ?, ?).

Autre activité symplectique à Paris:
- Séminaire Nantes-Orsay
- Doctorat Honoris Causa de Dusa McDuff : les 10 et 11 octobre à Paris 6.