### Symplectix 10 juin 2022

Location: IHP, room 01 in the morning, amphi Darboux in the afternoon.
The seminar will take place in presence, but will be broadcasted via zoom

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

10:45 Maksim Stokic (Tel Aviv)
C⁰ contact geometry of isotropic submanifolds.
Abstract: Homeomorphism is called contact if it can be written as C^0-limit of contactomorphisms. The contact version of Eliashberg-Gromov rigidity theorem states that smooth contact homeomorphisms preserve contact structure. Submanifold L of a contact manifold (Y,\xi) is called isotropic if \xi|_{TL}=0. Isotropic submanifolds of maximal dimension are called Legendrian, otherwise we call them subcritical isotropic.
In this talk, we will try to answer whether the isotropic property is preserved by contact homeomorphisms. It is expected that subcritical isotropic submanifolds are flexible, while we expect that Legendrians are rigid. We show that subcritical isotropic curves are flexible, and we give a new proof of the rigidity of Legendrians in dimension 3. Moreover, we provide a certain type of rigidity of Legendrians in higher dimensions.

14:00 Vukasin Stojisavljevic (Jussieu)
Coarse nodal count via topological persistence

Abstract: Given an eigenfunction of the Laplace-Beltrami operator on a closed Riemannian manifold, its nodal domains are connected components of the complement of its zero set. A version of Courant's nodal domain theorem states that the number of nodal domains can be bounded from above in terms of the corresponding eigenvalue. A well-known question in spectral geometry asks whether a similar statement holds for linear combinations of eigenfunctions. While a direct generalization in this direction fails to be true, a positive answer can be given for coarse nodal count, i.e. by ignoring small oscillations. The proof relies on the theory of persistence modules and barcodes combined with multiscale polynomial approximations. The aim of the talk is to explain this set of ideas, how they relate to the above mentioned question, as well as how they can be used to prove other coarse extensions of the Courant's nodal domain theorem. The talk is based on a joint work in progress with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.

15:45
Alex Takeda (IHES)   CANCELLED !

Autre activité symplectique à Paris:

- Séminaire Nantes-Orsay (à Orsay le 18 mars)
- Symplectic Zoominar (les vendredis hors symplectix à 15:15, heure de Paris)

### Symplectix 13 mai 2022

Lieu: IHP, salle 201
The seminar will take place in presence, but will be broadcasted via zoom

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

10:45 Basak Gurel (Florida)
Forced existence of periodic orbits in symplectic dynamics.

Abstract: In this talk, we will consider the phenomenon of “forced” existence of infinitely many periodic orbits from a broad perspective and discuss several results and open questions in a variety of settings. In particular, we will touch upon and sketch a relatively simple proof of a recent result of Egor Shelukhin, extending a celebrated theorem of Franks to higher dimensions. The talk is partially based on joint work with Erman Cineli and Viktor Ginzburg.

14:00 Noah Porcelli (Cambridge)
Homotopical Lagrangian monodromy
Abstract:
Given a Lagrangian submanifold L in a symplectic manifold X, a natural question to ask is: what diffeomorphisms f:L → L can arise as the restriction of a Hamiltonian diffeomorphism of X? Assuming L is relatively exact, we will extend results of Hu-Lalonde-Leclercq about the action of f on the homology of L, and deduce that f must be homotopic to the identity if L is a sphere or K(\pi, 1). The proof will use various moduli spaces of pseudoholomorphic curves as well as input from string topology. While motivated by HLL’s Floer-theoretic proof, we will not encounter any Floer theory..

15:45
Ilaria Di Dedda (Londres)
A symplectic interpretation of Auslander algebras of type A.
Abstract: The theme of this talk will be to build a bridge between two areas of mathematics: representation theory and symplectic geometry. Our objects of interest on the representation theoretical side (which I will define in this talk) are Auslander algebras of type A. This family of non-commutative algebras arises very naturally as endomorphism algebras of indecomposable modules of quivers of finite type. They were given a symplectic interpretation by Dyckerhoff-Jasso-Lekili, who proved the equivalence (as $A_{\infty}$-categories) between perfect derived categories of Auslander algebras of type A and certain partially wrapped Fukaya categories. We use their result to prove an equivalence between the categories in question and the Fukaya-Seidel categories of a certain family of Lefschetz fibrations. In this talk, we will observe this result in some key examples.

Prochaines séances: 10 juin (?, ?, ?)

Autre activité symplectique à Paris:

- Séminaire Nantes-Orsay (à Orsay le 18 mars)
- Symplectic Zoominar (les vendredis hors symplectix à 15:15, heure de Paris)

### Symplectix 1er avril 2022

Lieu: IHP, salle 201
The seminar will take place in presence, but will be broadcasted via zoom

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

10:45 Jonathan Bowden (Regensburg)
Open books, Bourgeois contact structures and their properties.
Abstract: Twenty years ago Frederic Bourgeois introduced a construction of contact structures on the product of any contact manifold M with a 2-torus given a choice of compatible open book, whose existence was proven by Giroux-Mohsen. In particular, this yielded contact structures on all odd-dimensional tori answering a question of Lutz from the 70’s. A systematic study of these contact manifolds was initiated by Lisi-Marinkovic-Niederkrüger and Gironella, the former asking several questions, which we address in this talk.
In particular, we show that if the initial contact manifold is 3-dimensional the resulting contact structure is tight, independent of the initial contact structure and choice of open book. Furthermore, we show that given ANY contact manifold one can always stabilise the open book so that the resulting contact structure is not strongly symplectically fillable. This then yields (many) examples of weakly but not strongly fillable contact structures in all dimensions. (joint work with F. Gironella and A. Moreno).

12:00 Henri Poincaré (Nancy)
From Floer theory to analysis situs.
Abstract
: I'll review some old but revolutionary ideas. I'll use beamer!

14:00 Abror Pirnapasov (Bochum)
Reeb orbits that force topological entropy
Abstract: A transverse link in a contact 3-manifold forces topological entropy if every Reeb flow possessing this link as a set of periodic orbits has positive topological entropy. We will explain how cylindrical contact homology on the complement of transverse links can be used to show that certain transverse links force topological entropy. As an application, we show that on every closed contact 3-manifold exists transverse knots that force topological entropy. We also generalize to the category of Reeb flows a beautiful result due to Denvir and Mackay, which says that if a Riemannian metric on the two-dimensional torus has a contractible closed geodesic, then its geodesic flow has positive topological entropy. All this is joint work with Marcelo R.R. Alves, Umberto L. Hryniewicz, and Pedro A.S. Salomão. BA.

15:45
Guillem Cazassus (Oxford)
Hopf algebras, equivariant Lagrangian Floer homology, and
cornered instanton theory.
Abstract: Let G be a compact Lie group acting on a symplectic manifold M in a Hamiltonian way. If L, L’ is a pair of Lagrangians in M, we show that the Floer complex CF(L,L’) is an A-infinity module over the Morse complex CM(G) (which has an A-infinity algebra structure involving the group multiplication). This permits to define several versions of equivariant Floer homology. This should also imply that the Fukaya categoy Fuk(M), in addition to its own A-infinity structure, is an A-infinity module over CM(G). These two structures should be packaged into a single one: CM(G) is an A-infinity bialgebra, and Fuk(M) is a module over it. In fact, CM(G) should have more structure, it should be a Hopf-infinity algebra, a strong-homotopy structure (still unclear to us) that should induce the Hopf algebra structure on H_*(G). In a certain sense, this refines a conjecture by Teleman.
Applied to some subsets of Huebschmann-Jeffrey’s extended moduli spaces introduced by Manolescu and Woodward, this construction should permit to define a cornered instanton theory analogous to Douglas-Lipshitz-Manolescu’s construction in Heegaard-Floer theory. This is work in progress, joint with Paul Kirk, Mike Miller-Eismeier and Wai-Kit Yeung.

Prochaines séances: 13 mai (I. Di Deda, N. Porcelli, ?), 10 juin (?, ?, ?)

Autre activité symplectique à Paris:

- Séminaire Nantes-Orsay (à Orsay le 18 mars)
- Symplectic Zoominar (les vendredis hors symplectix à 15:15, heure de Paris)