Author: Daniel Tsodikovich

Adi Dickstein (Tel Aviv University)

Topological quasi-states are special functionals on the algebra of continuous functions which are linear on single-generated subalgebras. They trace their origins to the von Neumann axioms of quantum mechanics. On symplectic surfaces, every topological quasi-state is symplectic, i.e., linear on Poisson-commutative subalgebras. We discuss the failure of this phenomenon in higher dimensions based on the study of symplectic embeddings of polydiscs. Furthermore, we introduce a Wasserstein-type metric on quasi-states and use it for quantitative constraints on symplectic quasi-states. The talk is based on a joint work with Frol Zapolsky.

Corentin Fierobe (University of Rome Tor Vergata)

Mathematical billiards in strictly convex domains with smooth boundaries provide tangible examples of twist maps on the cylinder, where the dynamics exhibit “almost integrable” behavior near the boundary. Building on this idea, Lazutkin demonstrated the existence of a Cantor set of positive measure, which contains zero, and within which the billiard maps have invariant curves associated with certain rotation numbers. These invariant curves evolve smoothly as the rotation number varies, in the Whitney sense. In this talk, I will present a generalization of this result for billiards with analytic boundaries, a joint work with Frank Trujillo and Vadim Kaloshin, motivated by recent advances from Carminati, Marmi, Sauzin, and Sorrentino. This extension shows that the Cantor set of rotation numbers can be continued into the complex plane, with the complex counterpart containing structures known as “diamonds.” This discovery offers new insights into length spectral rigidity.

Olga Bernardi (University of Padova)

Symplectic billiards were introduced by P. Albers and S. Tabachnikov in 2018 as a simple dynamical system where, opposed to Birkhoff billiards, the generating function is the area instead of the length. We first recall the main properties of symplectic billiards in strictly-convex domains. We proceed by presenting 
this recent result: if the phase-space is fully foliated by continuous invariant curves which are not null-homotopic, then the boundary of the billiard table is an ellipse.
 We finally discuss the state of the art 
both of integrability and of area spectral rigidity for these billiards. 
Joint works with Luca Baracco and Alessandra Nardi.