Symplectix 2 décembre 2022

 

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.

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..


14:00
Jean-Philippe Chassé (ETH Zurich)
The behavior of Hausdorff-converging sequences of Lagrangian submanifolds

Abstract:
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.


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)

Symplectix 18 novembre 2022

Location: IHP, room Darboux

The seminar will take place in presence, but will be broadcasted via zoom:
https://us02web.zoom.us/j/89445897442?pwd=QmVZWHBiM2Nwb09yUVJERnYrSHJrUT09


Exceptionnally and on account of the ANR Cosy meeting, we will have only one talk for this session of the symplectix seminar.


15:45 Dustin Connery-Grigg (Jussieu)
The geometry of Hamiltonian Floer theory on surfaces
Abstract:
In general, it is a difficult problem to understand how the Floer theory of a pair (H,J) relates to the qualitative dynamics  of the isotopy generated by H. In this talk I will present a novel definition of spectral invariants for Hamiltonian systems and explain how low-dimensional techniques can be used both to compute the most important of these in purely dynamical terms on surfaces, as well as to provide a bridge between Hamiltonian Floer theory and Le Calvez's theory of transverse foliations for dynamics on surfaces.

Next Symplectix:

Dec 2, Jan 6 ...

Other symplectic activity in Paris:

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

Symplectix 7 octobre 2022

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
Luca Asselle (Bochum)
Morse homology for the Hamiltonian action in cotangent bundles
Abstract: As first shown by Viterbo in a seminal paper from 1999, Floer homology for the Hamiltonian action functional for closed loops in the cotangent bundle of a closed smooth manifold is well-defined and in an essentially canonical way isomorphic to the loop space homology (in case of spin manifolds), provided the Hamiltonian is asymptotically quadratic in the fibre. In this talk I will show that, for such Hamiltonian actions, Morse homology is indeed well-defined in a direct way. The key idea is that the fibrewise linear structure of the cotangent bundle allows to split the Sobolev space setup for the loop space (according to the polarisation of the cotangent bundle as a family of Lagrangian fibres) such that the resulting Hilbert manifold of loops provides the Palais-Smale property for the action functional. The construction of Morse homology is then completed following the abstract work of Abbondandolo and Majer. As one expects, the resulting Morse homology is isomorphic to Floer homology. However, the Morse homology approach has potentially several advantages which will be discussed if time permits.


14:00
Denis Auroux (Harvard)
Fukaya categories of Landau-Ginzburg models and functoriality in mirror symmetry.
Abstract: The central topic of this talk will be Fukaya categories of Landau-Ginzburg models (i.e., symplectic fibrations over the complex plane). We will describe several natural functors relating these to other flavors of Fukaya categories, and the interpretation of these functors under mirror symmetry, where they correspond to inclusion and restriction functors between derived categories of coherent sheaves on a variety and a hypersurface inside it. The talk will be mostly
expository; the non-expository parts are joint work with Mohammed Abouzaid on one hand, and the thesis work of Maxim Jeffs on the other hand.


15:45 Yuichi Ike (Tokyo) 
Completeness of interleaving distance on Tamarkin category and C^0-symplectic geometry.
Abstract: The Tamarkin category is a sheaf category that can be used for studying non-displaceability problems in symplectic geometry. One can equip the category with a canonical interleaving-like distance to get quantitative information on Hamiltonian diffeomorphisms. In particular, the distance is stable with respect to Hamiltonian deformations of sheaves, which gives a sheaf-theoretic lower bound of displacement energy. In this talk, I will explain the completeness of the distance and its application to C^0-symplectic geometry. For that purpose, we also develop the Lusternik--Schnirelmann theory for sheaves. Joint work with Tomohiro Asano.


Next Symplectix:

Nov 18, Dec 2, Jan 6 ...

Other symplectic activity in Paris:

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