Persistence Homology in Symplectic and Contact Topology

Liste des Contributions

18. Welcome

12/06/2023 09:00

1. Interlevel persistence in Floer theory

Michael Usher

12/06/2023 09:30

The usual filtered Floer homology groups are formal analogues of the homologies of the sublevel sets of a Morse function on a manifold. In the Morse setting,
by instead considering interlevel sets (preimages of general intervals) one obtains an algebraic structure that is classified by a barcode that refines the usual
sublevel persistence barcode. I will describe an algebraic formalism...

2. Coarse topology, oscillations and persistence

Vukasin Stojisalvjevic

12/06/2023 11:00

Studying topology of the zero set of a polynomial map is a classical topic in algebraic geometry. One may attempt to extend this study to less regular objects, such as linear combinations of Laplace-Beltrami eigenfunctions or entire maps in several complex variables. A phenomenon which ultimately breaks the analogy is the existence of highly oscillatory behaviour on small scales. We will...

16. Sheaves for the completion of the space of Lagrangians

Stéphane Guillermou

12/06/2023 14:30

For a manifold $M$ it is known that we can associate a sheaf on $M\times \mathbb{R}$ to any exact Lagrangian submanifold of $T^*M$. The space of exact Lagrangians carries the Viterbo's metric and, following Humili`ere, we can consider its completion. We will see that the correspondence between sheaves and symplectic geometry extends to this framework and, moreover, natural notions defined...

4. Symplectic embeddings into convex toric domains (via Zoom)

Dusa McDuff

12/06/2023 16:00

This is a report on joint work with Cristofaro-Gardiner and Magill, that investigates symplectic embeddings into toric domains with irrational convex boundary.

5. A conjectural chain model for positive $S^1$-equivariant symplectic homology of star-shaped toric domains in $\mathbb{C}^2$ (via Zoom)

Kei Irie

13/06/2023 09:30

For any star-shaped toric domain in $\mathbb{C}^2$, we define a filtered chain complex which conjecturally computes positive $S^1$-equivariant symplectic homology of the domain.
Assuming this conjecture, we show that the sequence $(c^{\mathrm{GH}}_k(X)/k)_k$ has a limit as $k$ goes to $\infty$, where $c^{\mathrm{GH}}_k$ denotes the $k$-th Gutt-Hutchings capacity.

6. Contact Hamiltonian Floer homology and its applications

Jun Zhang

13/06/2023 11:00

In this talk, we will discuss a Floer-theoretic approach to studying Hamiltonian dynamics on contact manifolds, called contact Hamiltonian Floer homology. On the one hand, it provides a characterization of the rigidity of positive loops in the contactomorphism group which is a peculiar object often investigated in contact geometry. Mysteriously, our characterization highly relates to certain...

7. Sesquicuspidal curves

Kyler Siegel

13/06/2023 14:30

A classic question in algebraic geometry asks what are the possible singularities for a plane curve of a given degree and genus. This turns out to be closely connected with the theory of (stabilized) symplectic embeddings of ellipsoids. In this talk I will describe a construction (joint with D. McDuff) of new families of rational plane curves with desirable singularities. Key ingredients...

8. The Toda lattice, billiards and the Viterbo conjecture (via Zoom)

Vinicius Ramos

13/06/2023 16:00

The Toda lattice is one of the earliest examples of non-linear completely integrable systems. Under a large deformation, the Hamiltonian flow can be seen to converge to a billiard flow in a simplex. In the 1970s, action-angle coordinates were computed for the standard system using a non-canonical transformation and some spectral theory. In this talk, I will explain how to adapt these...

9. Spectral norm and conformally symplectic maps: two applications

Vincent Humilière

14/06/2023 09:30

I'll present two applications of the conformal invariance of the spectral norm: 1) a generalization of the Birkhoff attractor to higher dimension, 2) a weak version of the Nearby Lagrangian Conjecture. The completion of the space of Lagrangians with respect to the spectral norm plays a role in both cases.
This is joint work with Marie-Claude Arnaud and Claude Viterbo.

10. On the Hofer-rigidity of iterations and continuous Hamiltonian torsion

Egor Shelukhin

14/06/2023 11:00

We discuss a new rigidity phenomenon for iterations of Hamiltonian maps in the Hofer metric and in the C^0 metric. Joint work in progress with Nicholas Wilkins

19. Guided visit of the town of Albi

14/06/2023 15:30

20. Cocktail at the Hôtel de Ville

14/06/2023 18:00

11. Lagrangian link quasimorphisms and the non-simplicity of Hameomorphism group of surfaces

Cheuk Yu Mak

15/06/2023 09:30

In this talk, we will explain the construction of a sequence of homogeneous quasi-morphisms of the area-preserving homeomorphism group of the sphere using Lagrangian Floer theory for links. This sequence of quasi-morphisms has asymptotically vanishing defects, so it is asymptotically a homomorphism. It enables us to show that the Hameomorphism group is not the smallest normal subgroup of the...

12. Symplectic Barriers

Pazit Haim-Kislev

15/06/2023 11:00

In his seminal 2001 paper, Biran introduced the concept of Lagrangian Barriers, a symplectic rigidity phenomenon coming from obligatory intersections with Lagrangian submanifolds which doesn't come from mere topology.
One simple example is that when removing a Lagrangian plane from the four dimensional ball, the symplectic capacity shrinks down to half of the original capacity of the ball....

13. On the strong Arnold chord conjecture for convex contact forms

Jungsoo Kang

15/06/2023 14:30

Arnold conjectured that every closed Legendrian submanifold in the standard contact sphere $S^{2n-1}$ with a contact form has a Reeb chord. This was confirmed by K. Mohnke in 2001. In fact, Arnold originally conjectured that a Reeb chord with distinct endpoints exists. I will give a proof of this strong version of the conjecture for convex contact forms, namely contact forms on $S^{2n-1}$...

14. Completed signed barcodes and zeta functions for open domains

Michael Hutchings

15/06/2023 16:00

We describe some invariants of open domains related to bar codes coming from ECH and contact homology. These can be used for example to show that some open domains are not toric, and to distinguish some open toric domains which are difficult to distinguish otherwise.

15. The local strong Viterbo conjecture

Oliver Edtmair

16/06/2023 09:30

I will report on joint work in progress with Abbondandolo and Benedetti confirming the strong Viterbo conjecture for convex domains close to the ball in arbitrary dimension. The proof involves a quasi-invariant normal form for contact forms close to a Zoll one. I will explain how this normal form can be interpreted in terms of geodesics for symplectic Banach-Mazur type distances.

3. Elementary SFT Spectral Invariants And The Strong Closing Property

Julian Chaidez

16/06/2023 11:00

In this talk, I will discuss ESFT spectral gaps: a new, general class of spectral invariants for large class of stable Hamiltonian manifolds and their cobordisms, in any dimension. They are built using a min-max construction applied to J-holomorphic curves in symplectic field theory. I will explain their formal properties and provide a holomorphic curve "closing criterion" for the ESFT gaps...

17. Inverse reduction inequalities

Claude Viterbo

16/06/2023 14:00

Classically symplectic reduction yields an estimate from below for the spectral distance between Lagrangians (for example in a cotangent bundle) by the spectral distance between reductions.
We shall show here how using a result by Kiselev-Shelukhin using some enriched structure on barcodes, one can obtain upper bounds on the spectral distance from above and give a number of applications and...