### Symplectix 10 mars 2023

WARNING: the schedule changed after cancellation of Gabriel Rivière's talk

Location: IHP, amphi Darboux

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

10:45
Xiaohan Yan (Paris - Jussieu)
Divisor equations in quantum K-theory.
Abstract: Quantum K-theory studies a K-theoretic analogue of Gromov-Witten invariants defined as holomorphic Euler characteristics over the moduli spaces of stable maps. The K-theoretic GW invariants so defined satisfy similar axioms as their cohomological counterparts do, except the divisor equation. In genus-zero case, substitutes of the divisor equation in quantum K-theory have been proposed using the idea of toric fixed point localization and that of the so-called adelic characterization, both leading to interesting applications. For instance, explicit formulas of the J-functions of flag varieties can thus be justified, and the phenomenon named the quantum Serre duality be demonstrated. Higher-genus substitutes of the divisor equation remain however largely unknown.

14:00 Mélanie Theilliere (Luxembourg)
Le plan hyperbolique dans E^3.
Abstract: Depuis un résultat d'Hilbert-Effimov, nous savons que nous ne pouvons pas plonger isométriquement le plan hyperbolique dans l'espace euclidien de
dimension 3 de manière C^2. En revanche, le théorème de plongement
isométrique C^1 de Nash-Kuiper établit l'existence d'une infinité de tels
plongements. Dans cet exposé nous verrons la construction explicite
d'un plongement isométrique du disque de Poincaré, et nous donnerons
des résultats sur le "bord à l'infini" de ce type de plongement. Ce travail a été fait en collaboration avec l'équipe Hévéa.
https://www.concours-preuve-image.fr/linfini-trouve-toujours-son-chemin/

Next Symplectix:

Apr 7 (Benedetti, Maret, Marty) ...

Other symplectic activity in Paris:

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

### Symplectix 3 février 2023

Location: IHP, room 201

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

10:45
Nate Bottman (Bonn)
Constrainahedra and the Fukaya category of Lagrangian torus fibrations.
Abstract: I will begin by describing two pieces of the context for this talk: first, the symplectic (A-infinity,2)-category (Symp), which is the natural setting for building functors between Fukaya categories; and second, Lagrangian torus fibrations, which are the central geometric objects in SYZ mirror symmetry. Next, I will explain my construction with Daria Poliakova of a family of polytopes called constrainahedra, which we introduced in order to define the notion of a monoidal A-infinity category. Finally, I will describe work-in-progress that aims to equip the Fukaya category of a Lagrangian torus fibration with a monoidal A-infinity structure, which should be mirror to the tensor product of sheaves. This is based on past and ongoing work with Daria Poliakova and Mohammed Abouzaid, including arXiv:2208.14529 and arXiv:2210.11159.

14:00
Ivan Smith (Cambridge)
Morava K-theory and Hamiltonian loops.

Abstract:
I will discuss constraints on the symplectic topology of Hamiltonian fibrations with fibre a closed symplectic manifold. These constraints arise  from Floer homotopy theory’, by considering the fundamental classes of moduli spaces of holomorphic sections of the fibration in extraordinary cohomology theories.  This talk reports on joint work with Mohammed Abouzaid and Mark McLean.

15:45 Marco Mazzucchelli (Lyon)
C² structurally stable Riemannian geodesic flows of clsoed surfaces are Anosov.

Abstract: It is a celebrated claim of Poincaré that any positively curved Riemannian 2-sphere has a parabolic or elliptic closed geodesic (indeed, Poincaré even asserted the existed of a simple such closed geodesic, although this turned out to be wrong). This claim has been confirmed generically by Contreras and Oliveira, without requirements on the curvature: a C² generic Riemannian metric on the 2-sphere has an elliptic closed geodesic. In this talk, I will present a generalization of this result to arbitrary closed surfaces: a C² generic Riemannian metric on a closed surface has either an elliptic closed geodesic or an Anosov geodesic flow. A consequence of this statement is a confirmation of the C² stability conjecture for Riemannian geodesic flows of closed surfaces: any such geodesic flow that is C² structurally stable within the class of Riemannian geodesic flows must be Anosov. The proof is based on a new characterization of Anosov Reeb flows of closed contact 3-manifolds. This is joint work with Gonzalo Contreras.

Next Symplectix:

Mar 10 (Rivière, Theilliere, Yan), Apr 7 (Benedetti (TBC), Maret, Marty) ...

Other symplectic activity in Paris:

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

### Symplectix 6 janvier 2023

Location: IHP, room 201

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

10:45
Julian Chaidez (Princeton)
Legendrian ECH.

Abstract: Embedded contact homology (ECH) is a flavor of SFT for contact 3-manifolds that has been very successful at resolving open problems in Reeb dynamics, such as the Weinstein conjecture. Many versions of SFT admit generalizations to relative theories involving Legendrian (or Lagrangian) boundary conditions. The goal of this talk will be to explain the construction of the Legendrian version of ECH and discuss various potential applications and open problems. Our construction builds on several ideas present in previous works, such as Colin-Ghiggini-Honda's isomorphism between HF and ECH. This is joint work with Oliver Edtmair, Luya Wang, Yuan Yao and Ziwen Zhao.

14:00
Thomas Massoni (Princeton)
Anosov flows, non-Weinstein Liouville domains and their wrapped Fukaya categories.

Abstract:
While Weinstein domains and their symplectic invariants have been extensively studied over the last 30 years, little is known about non-Weinstein Liouville domains. We present a construction in dimension four based on Anosov flows on three-manifolds. The symplectic invariants of these `Anosov Liouville domains'' constitute new invariants of Anosov flows. The algebraic structure of their wrapped Fukaya categories is in stark contrast with the Weinstein case. We focus on a subcategory $\mathcal{W}_0$ of the wrapped Fukaya category whose objects are in bijection with the simple closed orbits of the flow. Surprisingly, $\mathcal{W}_0$ is not homologically smooth, as it is not finitely split-generated in a maximal way. This talk is mostly based on joint work arXiv:2211.07453 with Kai Cieliebak, Oleg Lazarev and Agustin Moreno.

15:45 Oliver Edtmair (Berkeley)
Disk-like surfaces of section and symplectic embeddings.
Abstract: Symplectic embedding problems, i.e. the question whether one symplectic manifold embeds into another, are of central importance in symplectic geometry. Such problems are intimately related to Hamiltonian dynamics and this relationship has been used to construct a plethora of obstructions to symplectic embeddings. Going in the opposite direction, I will discuss how disk-like global surfaces of section, a concept from dynamics, can be used to construct symplectic embeddings. This yields partial progress towards Viterbo’s conjecture on symplectic capacities of convex domains: In dimension four, the cylindrical embedding capacity agrees with the minimal action of an unknotted Reeb orbit.

Next Symplectix:

Feb 3 (Bottman, Smith, Mazzucchelli), Mar 6 (Theillere, ?, ?)...

Other symplectic activity in Paris:

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