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.

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.

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.

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.

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.

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.

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.

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.