Séance du 15 septembre 2017

Lieu: IHP, salle 201

11:00  Umut Varolgunes (MIT)
Mayer-Vietoris sequence for relative symplectic cohomology.
Abstract: I will first recall the definition of an invariant that assigns to any compact subset K of a closed symplectic manifold M a module SH_M(K) over the Novikov ring. I will go over the case of M=two sphere to illustrate various points about the invariant. Finally, I will state the Mayer-Vietoris property and explain under what conditions it holds.


14:15 Doris Hein (Freiburg)  
Local invariant Morse theory and applications in Hamiltonian dynamics. 
Abstract:  Local homology is a useful tool to study periodic orbits. For example, the key to the existence of infinitely many periodic orbits of Hamiltonian systems are properties of the local Floer homology of one special orbit. I will discuss a discrete version of this invariant constructed using local invariant Morse homology of a discrete action function. The construction is very geometric and relies on a hands-on description of invariant local Morse homology. The resulting local homology can be interpreted as an invariant of germs of Hamiltonian systems or of closed Reeb orbits. It has properties similar to those of local Floer homology in the symplectic setting and probably similar applications in dynamics.

16:00  Noémie Legout (Orsay)   

Products in Floer theory for Lagrangian cobordisms.
 Abstract: Chantraine, Dimitroglou-Rizell, Ghiggini and Golovko have defined the Cthulhu complex, a Floer complex for Lagrangian cobordisms. The acyclicity of this complex permits to rely the Legendrian contact homologies of the positive and negative Legendrian ends of the cobordism and the singular homology of the cobordism through various long exact sequences. In this talk, I will recall briefly the definitions of Legendrian contact homology and Cthulhu complex, and I will explain how to define products on a subcomplex of the Cthulhu complex..

Prochaines séances: 6/10 (Gutt, Pedroza, Vaintrob), 10/11 (Näf, Siefring, ? ), 2/12 (? , ? , ? )

Autre activité symplectique à Paris:
- Séminaire Nantes-Orsay,

Séance du 2 juin 2017

Lieu: IHP, salle 201

11:00  Honghao Gao (Northwestern & Jussieu) 
Augmentations and sheaves for knot conormals.
Abstract: Knot invariants can be defined using Legendrian isotopy invariants of the knot conormal. There are two types of invariants raised in this way: one is the knot contact differential graded algebra together with augmentations associated to this dga, and the other one is the category of simple sheaves microsupported along the knot conormal. Nadler-Zaslow correspondence suggests a connection between the two types of invariants. Moreover, augmentations specialized to “Q=1” have been understood through KCH representations.
I will present a classification result of simple sheaves, and relate it to KCH representations and two-variable augmentation polynomials. I will also present a Radon transform for sheaf categories, and explain how it corresponds to the specialization of Q on the sheaf side.


14:15 Maia Fraser (Ottawa)   
Generating function-based capacities, old and new, and contact non-squeezing.  
Viterbo’s symplectic capacity of domains in R^{2n} and Sandon’s contact capacity of domains in R^{2n} × S^1 can be seen as persistences of certain homology classes in the persistence module formed by generating function (GF) homology groups. While Sandon’s capacity c_S(−) allows to re-prove non-squeezing of any pre-quantized ball B(R) × S^1 with integral R into itself (originally due to Eliashberg-Kim-Polterovich 2006), by introducing filtration-decreasing morphisms between GF homology groups one can set up a functor from a sub-category of the poset D × Z to Vect, where D is the category of bounded domains with inclusion. Persistences in this persistence module yield a sequence m_\ell(−), \ell ∈ N of integer-valued contact capacities for pre-quantized balls, such that m_1 is related to c_S and higher m_\ell allow to re-prove non-squeezing via contact isotopies of B(R) × S^1 into itself for any R>1 (originally due to Chiu 2014). I will sketch the construction of these new GF capacities and revisit old ones along the way.

16:00  Claude Viterbo (ENS)    
Quantification des Lagrangiennes par le complexe de Floer.
 On montre comment quantifier des lagrangiennes du cotangent en utilisant le complexe de Floer. On en déduit par ailleurs un certain nombre de résultats liant l'homologie de Floer et faisceaux.

Prochaines séances: après l'été...

Autre activité symplectique à Paris:
- Séminaire Nantes-Orsay,