**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.