Workshop in Geometry and Dynamics

Valentina Franceschi

Magnetic fields and improved Hardy inequalities in the Heisenberg group

The aim of this seminar is to discuss magnetic fields on the Heisenberg group and their influence on the Heisenberg sub-Laplacian.

We first give an overview of the Euclidean situation, discussing improved Hardy inequalities for the magnetic Euclidean Laplacian, following Laptev and Weidl standard results.

We then pass to describe magnetic fields in the Heisenberg group and the associated Hardy-type inequalities.

Instrumental for our discussion is a Hardy-type inequality for the Folland-Stein operator, that we show to hold true and has an interest on its own.

Based on a joint work with Biagio Cassano, David Krejcirik, Dario Prandi.

Davide Barilari

Synthetic curvature bounds and sub-Riemannian geometry.

The theory of CD(K,N) spaces, synthetic version of Ricci curvature lower bounds proposed by Lott-Sturm-Villani, has been shown not to directly apply to sub-Riemannian geometry. Nonetheless, using optimal transport, entropy inequalities yielding new versions of Brunn-Minkowski inequalities have been proved to be true in the case of the Heisenberg group and more in general in ideal sub-Riemannian manifold. For this reason, in his 2017 Bourbaki seminar, Villani conjectured there is a hope for a ``grande unification'' of Riemannian, Finslerian and Sub-Riemannian synthetic Ricci lower bounds.

In this talk I will report on these results and present a recent new approach to the problem in collaboration with Andrea Mondino and Luca Rizzi.

Luca Asselle

Exotic Zoll magnetic systems on the two-torus via the Nash-Moser IFT.

In this talk we show that there is an infinite dimensional space of magnetic systems on the two-torus for which every magnetic geodesic with speed 1 is periodic. Such a result is a magnetic analogue of Guillemin‘s theorem for Zoll metrics on the two-sphere, and its proof is (exactly as in Guillemin) based on a Nash-Moser type implicit function theorem. We will briefly discuss the analytical setup of the problem and explain why the classical IFT fails. We then address some of the points needed in order to apply the Nash-Moser IFT. From a technical point of view, the analysis is in this case significantly more delicate due to the fact that the magnetic flow is non-reversible. In fact, for the applicability of the Nash-Moser IFT a key role is played by the symmetry properties of Bessel‘s functions. This is joint work with Gabriele Benedetti and Massimiliano Berti.

Richard Montgomery

2 questions for the 3-body problem.

The classical gravitational 3-body problem is far from dead. I will state two open questions in the area, hint at another, and sketch my trajectory from gauge theory to the 3-body problem,passing through the falling cat and subRiemannian geometry. Time most likely not permitting, I will present progress on these two problems.

Alvaro del Pino

On sub-Riemannian geodesics and billiard trajectories.

Given a 3-dimensional contact manifold, one can endow the contact structure with a metric; we call this a sub-Riemannian structure. Much like in the Riemannian case, one can then try to determine the curves (necessarily tangent to the contact structure) that minimise length. We call these sub-Riemannian geodesics. When the manifold has boundary, there is a natural reflection law that determines how geodesics bounce when they hit the boundary. This allows us to talk about sub-Riemannian billiard trajectories. In this talk I will discuss a few examples, as well as various open problems in this direction.

José Pedro Gaivão

Generic properties of multidimensional billiards.

In this talk we discuss generic properties of billiards inside strictly smooth convex bodies. In particular, we show that in a C2-open and dense set of strictly convex bodies, the billiard has positive topological entropy. This is a joint work with Mário Bessa, João Lopes Dias, Gianluigi Del Magno and Maria Joana Torres.

Gabriele Benedetti

The dynamics of strong magnetic fields on surfaces.

In this talk, we study the motion of a charged particle on a closed surface under the effect of a strong magnetic field. By means of a Hamiltonian normal form, we will construct periodic orbits and trapping regions for the particle, and prove a Bertrand-type theorem on the rigidity of magnetic fields all of whose orbits are periodic. This is joint work with Luca Asselle.

Pazit Haim Kislev

Symplectic capacities of p-products.

We discuss the behavior of action minimizing closed characteristics on the boundary of symplectic p-products of two convex domains, which is a convex body that lives between their free sum and their Cartesian product.

One application, by using a "tensor power trick", is to show that it is enough to prove Viterbo's volume-capacity conjecture in the asymptotic regime when the dimension is sent to infinity.

In addition, we introduce a conjecture about higher order capacities of p-products and show that if it holds then there are no non-trivial p-decompositions of the symplectic ball.

Gabriel Paternain

Resonant forms at zero for dissipative Anosov flows.

The Ruelle zeta function is a natural function associated to the periods of closed orbits of an Anosov flow, and it is known to have a meromorphic extension to the whole complex plane. The order of vanishing of the Ruelle zeta function at zero is expected to carry interesting topological and dynamical information and can be computed in terms of certain resonant spaces of differential forms for the action of the Lie derivative on suitable spaces with anisotropic regularity.

In this talk I will explain how to compute these resonant spaces for any transitive Anosov flow in 3D, with particular emphasis in the dissipative case, that is, when the flow

does not preserve any absolutely continuous measure. A prototype example is given by the geodesic flow of a Weyl connection and we shall see that for such a flow the order of vanishing drops by 1 in relation to the usual geodesic flow due to the Sinai-Ruelle-Bowen measure having non-zero winding cycle. This is joint work with Mihajlo Cekić.