We will present an analog of Santaló point for the area of the boundary of a convex body in a normed space, and explain why they exist in general, and under which conditions they are unique. We will also provide a characterization of such a point when the normed space is Minkowski. This is joint work with G. Solanes and K. Tzanev.

On complete, non-compact Riemannian manifolds the possibilities for the behaviour of a geodesic with respect to infinity leads to a variety of interesting questions. A geodesic can be bounded or proper or oscillating. Moreover, and particularly in dimension two, the injectivity of geodesics is of interest. After a short and incomplete review of the history of such problems (that starts in the 19th century), the talk will concentrate on proper, injective geodesics on complete planes. S.Cohn-Vossen asked in 1936 for the existence of at least one such geodesic. This was answered affirmatively by the speaker (1981), when he also asked for the minimal number \(n\) of such geodesics on a complete plane. The talk will present an example of a complete plane with only two injective geodesics (joint work with S.Suhr), so \(n\) is either one or two. We conjecture that actually \(n=2\).

On a symplectic manifold endowed with a Lagrangian bundle and for conformally symplectic dynamics, the asymptotic Maslov index for a positive orbit is for the angles what the Lyapunov exponent is for the norms. In a joint work with Marie-Claude Arnaud and Valentine Roos, on a cotangent bundle and for a conformally symplectic dynamics, we show the existence of invariant measure with vanishing asymptotic Maslov index. If moreover the dynamics twists the vertical, then we prove that every Lagrangian submanifold \(L\) Hamiltonianly isotopic to a graph exhibits a point \(x\in L\) of zero index.

The purpose of this talk is to discuss some new sufficient conditions for the existence of rational global surfaces of section (Birkhoff sections) for Reeb flows in dimension three. Our conditions provide perturbative existence statements since they are C^infty generic. They hold, in particular, when the contact form is non-degenerate and the Liouville measure can be approximated by periodic orbits, but we will also explain that these conditions can be checked for certain families of explicit (possibly degenerate) Reeb flows. All results are joint work with Colin, Dehornoy and Rechtman.

The purpose of the talk will be to discuss the relation between positive and non-negative Legendrian and contact isotopies.

A strict contactomorphism is a diffeomorphism of a co-oriented contact manifold \((M,\alpha)\) that preserves the contact form \(\alpha\). Motivated by a problem on symplectic embeddings, we study the homotopy type of the group of strict contactomorphisms of a closed contact manifold. While this looks hard in general, computations are possible for generic contact forms, and in special situations as for boundaries of toric domains and for cosphere bundles. This is work in slow progress joint with Joé Brendel.