Author: Corentin Fierobe

Speaker. Daniel Tsodikovich
Date. 05.12.23 at 14:30

Abstract. The Blaschke-Santalo inequality is a classical inequality in convex geometry. This inequality is about the product of the volumes of a convex body and its dual. In this talk we investigate an analogue of this inequality, where the volume is replaced with the length of the shortest billiard trajectory. We focus on the two dimensional case. We will describe what the analogue of the “Santalo point” is in this setting, show an analogue of the inequality itself, and discuss maximizers in classes of polygons.

Speaker. Alberto Abbondandolo
Date. 24.10.23 at 14:30

Abstract. An old open question in symplectic geometry asks whether all normalised symplectic capacities coincide on convex bodies in the standard symplectic vector space. I will show that this question has a positive answer for smooth convex bodies which are C^2-close to a Euclidean ball. This is related to the question of existence of minimising geodesics in the space of contact forms on a closed contact manifold equipped with a Banach-Mazur-like metric. The talk is based on recent joint work with Gabriele Benedetti and Oliver Edtmair.

Speaker. Frank Trujillo
Date. 08.06.22 at 2:00 pm

Abstract. The classical KAM theory establishes the persistence, under sufficiently small perturbations, of most of the n-dimensional invariant tori for non-degenerate integrable Hamiltonians with n degrees of freedom. The surviving tori are those carrying a quasi-periodic motion by a Diophantine vector and, in particular, their restricted dynamics is minimal. On the other hand, such systems also admit n-dimensional invariant tori whose restricted dynamics is not minimal. These tori, which we call resonant, are foliated by lower dimensional invariant tori, that is, by invariant tori whose dimension is strictly smaller than the number of degrees of freedom of the system. The codimension of these invariant subtori (with respect to the resonant torus), is called the number of resonances of the resonant torus.
In this talk, I will present a criterion for the existence of at least one lower dimensional invariant torus, associated to a resonant invariant torus (with any number of resonances) in the unperturbed system, for a class of near-integrable non-convex Hamiltonians.

Speaker. Amir Vig
Date. 09.05.22 at 2:00 pm

Abstract. A classical inverse problem in mathematical physics is to determine the shape of a membrane from the resonant frequencies at which it vibrates. The problem is very much still open for smooth, strictly convex planar domains and in this case, there is a natural dynamical analogue: can one determine the shape of such a domain from the lengths of periodic billiard trajectories? In fact, these problems are related by the Poisson relation and generalized trace formulae which give asymptotic expansions for the wave trace (a Laplace spectral invariant) near the length of a periodic billiard trajectory. In this talk, I will discuss recent work on both types of inverse problems and describe an ongoing project to further elucidate the connection between them.hism be perturbed in the real analytic topology so that its BNF is convergent? Can such a diffeomorphism be perturbed so that it becomes integrable in a neighborhood of the origin?

Speaker. Raphaël Krikorian
Date. 09.05.22 at 3:30 pm

Abstract. To any symplectic real analytic diffeomorphisms of the 2-dimensional disk (or annulus) admitting the origin as a non resonant fixed point one can associate a formal series, the Birkhoff Normal Form (BNF), which is invariant by (formal) conjugations. One can prove that in general this formal series is divergent. I shall address in this talk the following questions: does the convergence of the BNF imply integrability of the diffeomorphism in a neighborhood of the origin? Can such a diffeomorphism be perturbed in the real analytic topology so that its BNF is convergent? Can such a diffeomorphism be perturbed so that it becomes integrable in a neighborhood of the origin?

March 1, 2022