Location: IHP, room 201

The seminar will take place in presence, but will be broadcasted via** **zoom:

https://us02web.zoom.us/j/89445897442?pwd=QmVZWHBiM2Nwb09yUVJERnYrSHJrUT09

**10:45** Cyril Falcon (Orsay)

Generating family homologies of Legendrian submanifolds and moduli spaces of gradient staircases.

Generating family homologies of Legendrian submanifolds and moduli spaces of gradient staircases.

*Abstract:*To overcome substantial technical difficulties arising in the practical use of generating families to explore the diversity of Legendrian submanifolds in contact topology, Henry and Rutherford introduce in a 2013 paper a singular perturbation of the gradient flow, used to define the homology of the generating family. They conjecture that when finite, the moduli spaces used to define the boundary operator of this homology are in one-to-one correspondence with some moduli spaces of staircase trajectories.

In this talk, I want to sketch the first step towards a complete proof by compactness-gluing of this conjecture. More precisely, I want to explain that in the Henry and Rutherford adiabatic limit, genuine gradient trajectories "accumulate" on gradient staircases chains, provided some geometric assumptions and transversality conditions are met. Then, if time permits, I would also like to carry several explicit generating family homology computations with gradient staircases and raise some conjectures from them.

This talk is based on my thesis work under the supervision of F. Bourgeois..

**Jean-Philippe Chassé (ETH Zurich)**

14:00

14:00

The behavior of Hausdorff-converging sequences of Lagrangian submanifolds

The behavior of Hausdorff-converging sequences of Lagrangian submanifolds

*A given collection of nice Lagrangian submanifolds of a symplectic manifold can be equipped with many symplectically-relevant metrics, e.g. the Lagrangian Hofer metric, the spectral metric, and the shadow metrics. However, understanding how these metrics are related to more set-theoretical measurements has proven to be quite hard. After presenting some recent results relating—when uniform Riemannian bounds are present—convergence in these metrics to Hausdorff-convergence, I will explain what such Hausdorff-convergence actually entails. This will be done by studying general sequences of immersions respecting Riemannian bounds at first, but the addition of the Lagrangian condition will cause some unique rigidity phenomena. Furthermore, I will explain how we can get a Lagrangian generalization of results of Hofer and Viterbo on simultaneous C⁰ and Hofer/spectral limits from these previous results. This generalization holds even when no Riemannian bounds are imposed.*

Abstract:

Abstract:

**15:45**Alex Takeda (IHES)

**Smooth Calabi-Yau structures and loop spaces.**

*Abstract*: In this talk I will discuss the relation between Poincaré duality for finite CW complexes, and a duality structure called smooth Calabi-Yau structure. It is well-known that the chain algebra of the based loop space of a Poincaré duality space carries this structure; however, that proof is rather non-local and uses topological group models for the spaces involved. I will explain how to understand this relation in a more local manner, replacing this algebra by an equivalent dg category coming from a simplicial set structure. This gives explicit chain-level formulas for the structures involved. Time allowing I will explain how to use this chain-level data to solve a certain Maurer-Cartan problem on a space of noncommutative polyvector fields. Part of this work is joint with M. Kontsevich and Y. Vlassopoulos.

**Next Symplectix:**

Jan 6, Fev 3 ...

**Other symplectic activity in Paris:**

**-**Séminaire Nantes-Orsay

- Symplectic Zoominar (every Fridays except Symplectix' Fridays at 15:15, Paris time)