Two days in Symplectic Dynamics
École normale supérieure de Lyon, France
September 29-30, 2022

The meeting will take place at the Site Jacques Monod of the École normale supérieure de Lyon, in the following lecture rooms:
• Thursday morning and Friday: Salle de thèses (ground floor)
• Thursday afternoon: Amphi A (4th floor)
Schedule

Thursday, September 29th
Friday, September 30th

Abstracts
• Lev Buhovsky - Groups of area-preserving homeomorphisms, spectral estimators, and Sikorav's trick
The celebrated Fathi question asked about the simplicity of the group of Hamiltonian homeomorphisms of a symplectic surface. The recent solution of the question introduced the group of finite energy Hamiltonian homeomorphisms which was shown to be a non-trivial normal subgroup, thus giving the negative answer to the question. That group of finite energy Hamiltonian homeomorphisms contains in itself the group of Oh-Müller Hamiltonian homeomorphisms. In my talk I will try to explain how one can compare between these groups and show that their quotient is large from the perspective of Hofer's geometry.

• Jean Gutt - Symplectic convexity? (an ongoing story...)
What is the symplectic analogue of being convex? We shall present different ideas to approach this question. Along the way, we shall present recent joint results with J. Dardennes and J. Zhang on monotone toric domains non-symplectomorphic to convex domains and with M. Pereira and V. Ramos on cube-normalized capacities.

• Patrice Le Calvez - The Calabi invariant of an irrational pseudo rotation; a finite dimensional approach
Some years ago, Michael Hutchings proved that the Calabi invariant of an irrational pseudo rotation of the disk is equal to its rotation number. The proof uses Embedded Contact Homology theory. We give a proof that uses finite dimensional arguments, related to the dynamical study of the gradient field of a generating function.

• Leonardo Macarini - Symmetric closed Reeb orbits on the standard contact sphere
A long standing conjecture in Hamiltonian Dynamics states that every contact form on the standard contact sphere $$S^{2n+1}$$ has at least $$n+1$$ simple periodic Reeb orbits. In this talk, I will consider a refinement of this problem when the contact form has a suitable symmetry and we ask if there are at least $$n+1$$ simple symmetric periodic orbits. We show that there is at least one symmetric periodic orbit for any contact form and at least two symmetric closed orbits whenever the contact form is dynamically convex. This is ongoing joint work with Miguel Abreu and Hui Liu.

• Eva Miranda - Two sides of a mirror
Etnyre and Ghrist unveiled a mirror between contact geometry and fluid dynamics reflecting Reeb vector fields as Beltrami vector fields. I'll present several applications of this mirror including the detection of escape trajectories (for which I'll need to extend the mirror to a singular set-up). This talk is based on joint works (some of them ongoing) with Robert Cardona, Josep Fontana, Cédric Oms, and Daniel Peralta-Salas.

• Daniel Peralta Salas - Hamiltonian dynamics of Gaussian random potentials
How is the dynamics of a natural Hamiltonian system (on the cotangent bundle of a compact manifold) for a generic potential? Is it typically nonintegrable or ergodic? In this lecture I will present this problem, which is widely open, and the probabilistic approach that we have developed to shed some light on it. Key to our construction is the concept of Gaussian random potential, which is substantially inspired by the celebrated theory of Nazarov and Sodin in the context of random monochromatic waves. This is based on joint work with A. Enciso and A. Romaniega.

• Leonid Polterovich - Lorentzian geometry on contactomorphisms, or why flexibility is expensive?
I'll discuss the motto "flexibility requires complexity" in the case study of a biinvariant Lorentz-Finsler geometry on contactomorphism groups. Joint with Alberto Abbondandolo and Gabriele Benedetti.

• Sobhan Seyfaddini - Comparisons between the $$C^0$$ and $$\gamma$$ metrics and applications
We will discuss the connections between the $$C^0$$ distance and the $$\gamma$$ distance and will see some of the ensuing applications. This is partially based on joint work with Dusan Jokscimovic.