Séance du 6 octobre 2017

Lieu: IHP, salle 201

11:00  Dmitry Vaintrob (IAS, Princeton)
Trivializing the circle action on the higher genus surface operad.
Abstract: I will introduce topological operads and their model structure and explain a homotopy equivalence (from recent work of the author with Alexandru Oancea) between two operads which is expected to be part of a hypothetical "higher genus mirror symmetry”. 
    Background: The original appearance of mirror symmetry in mathematics came from an observed numerical equality between Hodge numbers of a certain projective variety and counts of holomorphic spheres on its "mirror" considered as a symplectic variety. Maxim Kontsevich conjectured that this numerical equality has roots in an equivalence of derived categories between the category of coherent sheaves on one side and the "Fukaya category" associated to the symplectic structure on the other. Passage from the categorical equivalence to the numerical equivalence consists of taking dimensions of the Hochschild homology (a derived invariant) of the two categories. In particular, one expects (in nice situations) that the Hochschild homology of the Fukaya category is described by the homology of the moduli space of genus-zero curves. Now the homology of the space of genus-zero curves has naturally operations indexed by the (shifted) homology of Deligne-Mumford compactified moduli spaces $\mathcal{M}_{0, n}$, while the homology of the Fukaya category has action (using a dualizability property) by the (shifted) homology of moduli spaces of framed little disks (a very different algebraic structure). The disparity between these two actions can be explained neatly by a result from the homotopy theory of operads (first proven by Gabriel Drummond-Cole) using the triviality of a certain circle action. I will sketch our proof of an analogous result in higher genus.
14:15 Andrés Pedroza (Colima, Mexique) 
Lagrangian submanifolds in the one-point blow up of CP². 
Abstract: We will show how a Lagrangian submanifold in CP² that is Hamiltonian isotopic to RP², lifts to a Lagrangian submanifold in the symplectic one-point blow up of CP² such that is no longer Hamiltonian isotopic to the lift of RP². We show this by computing the Lagrangian Floer homology of the pair of Lagrangian submanifolds in the symplectic blow up in terms of the Lagrangian Floer homology of the pair in CP².

16:00  Jean Gutt (Cologne)  

Knotted symplectic embeddings.
 Abstract: I will discuss a joint result with Mike Usher, showing that many toric domains X in the 4-dimensional euclidean space admit symplectic embeddings f into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes f(X) to X.

Prochaines séances: 10/11 (K. Näf, R. Siefring), 1/12 (P. Biran, ? , ? )

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