An old open question in symplectic dynamics asks whether all normalized symplectic capacities coincide on convex domains. I will discuss this question and show that the answer is positive if we restrict the attention to domains which are close enough to a ball. The proof is based on a “quasi-invariant” normal form in Reeb dynamics, which has also implications about geodesics in the space of contact forms equipped with a Banach-Mazur pseudo-metric. This talk is based on a joined work with Gabriele Benedetti and Oliver Edtmair.[-]

The spectral norm is an important invariant of a Hamiltonian diffeomorphism and its properties have recently found numerous nontrivial applications to dynamics. We will explore the behavior of the spectral norm under iterations of a Hamiltonian diffeomorphism and some related phenomena. This feature is closely related to the existence of invariant sets (the Le Calvez–Yoccoz type theorems), Hamiltonian pseudo-rotations, rigidity and the strong closing lemma. The talk is based on joint work with Erman Çineli and Viktor Ginzburg.[-]