Contributed Talks

Abstract: We propose a theory "a la Conley" for cone fields using a notion of relaxed orbits based on cone enlargements, in the spirit of space time geometry. We work in the setting of closed (or equivalently semi-continuous) cone fields with singularities. This setting contains (for questions which are parametrization independent such as the existence of Lyapounov functions) the case of continuous vector-fields on manifolds, of differential inclusions, of Lorentzian metrics, and of continuous cone fields. We generalize to this setting the equivalence between stable causality and the existence of temporal functions. We also generalize the equivalence between global hyperbolicity and the existence of a steep temporal functions.

Abstract: Birkhoff's conjecture states that the only integrable billiards in the plane are ellipses. I am going to discuss some results and questions motivated by this conjecture.

Abstract : We study the time-periodic version of Evans approach to weak KAM theory. Evans minimization problem is equivalent to a first oder mean field game system. For the mechanical Hamiltonian we prove the existence of smooth solutions. We introduce the corresponding effective Lagrangian and Hamiltonian and prove that they are smooth. We also consider the limiting behavior of the effective Lagrangian and Hamiltonian, Mather measures and minimizers.

Abstract:We show that a smooth Lagrangian, minimizing torus that is invariant by the geodesic flow of a Riemannian metric in the torus is a graph of the canonical projection provided that every point in the Lagrangian torus is nonwandering. The graph problem for Lagrangian minimizing two dimensional tori was completely solved by Bialy and Polterovich in the 1980's and since then very llttle progress has been made in higher dimensions. This contrasts with the great development of the graph problem for Lagrangian invariant tori homologous to the zero section since the works of Viterbo and Polterovich in the early 1990's. The result is part of a joint research project with Mario Jorge Carneiro.

Abstract: In optimal control, minimizing trajectories are projections on the ambient manifold of Hamiltonian curves on the cotangent bundle (“extremal curves”). These curves may be smooth and we report on results on the geodesic flow of almost-Riemannian metrics on the 2-sphere. Such metrics have singularities that can be suitably dealt with in a Hamiltonian framework. We show in particular that their caustics are given in terms of a billiard in the Poincare disk. In general though, the relevant Hamiltonian is only C^0 and minimizing trajectories are projections of broken curves (Lipschitz but not C^1). An important case for the control of mechanical systems is the case of two competing Hamiltonians. Under suitable assumptions, neighbouring extremals are all broken and there is still a good notion of caustic. A more subtle situation arises for time minimization as neighbouring Hamiltonian curves of a broken extremal may be smooth or broken. In a well-chosen blow-up, the singularity of the extremal can be interpreted as a heteroclinic connection between two hyperbolic equilibria, resulting in a logarithmic singularity of the Hamiltonian flow.

Abstract: We study convex caustics in Minkowski billiards. We show that for the Euclidean billiard dynamics in a planar smooth centrally symmetric and strictly convex body K, for every convex caustic which K possesses, the "dual" billiard dynamics in which the table is the Euclidean unit disk and the geometry that governs the motion is induced by the body K, possesses a dual convex caustic. Such a pair of caustics is dual in a strong sense, and in particular they have the same perimeter, Lazutkin parameter (both measured with respect to the corresponding geometries), and rotation number. We show moreover that for general Minkowski billiards this phenomenon fails, and one can construct a smooth caustic in a Minkowski billiard table which possesses no dual convex caustic.

Abstract: Monotone curves are special curves in the Lagrangian Grassmanian that arise very naturally in the context of optimal control problems. For example, they can be used to construct canonical lifts of discontinuous curves to the universal cover of the Lagrangian Grassmanian. In a recent work with A. A. Agrachev we have used them to prove a general theorem in optimal control that generalizes the Morse theorem in the classical calculus of variations. This theorem states that the Morse index of the Hessian of the functional can be efficiently computed using the Maslov index of a curve in the Lagrangian Grassmanian called Jacobi curve.

Abstract: We show that symplectically embedded (−1)-tori give rise to certain elements in the symplectic mapping class group of 4-manifolds. An example is given where such elements are proved to be of infinite order.

Abstract: In this work we connect Poisson and near-symplectic geometry by showing that there are two almost regular Poisson structures induced by a near-symplectic 2n-manifold. The first structure is of maximal rank 2n and vanishes on a codimension-2 subspace. The second one is log-f symplectic of maximal rank 2n−2. We then compute the Poisson cohomology of the former structure in dimension 4, showing that it is finite and depends on the modular class. We also determine the cohomology of a different Poisson structure on smooth 4-manifolds, the one associated to broken Lefschetz fibrations. This completes the cohomology of the possible degeneracies of singular Poisson structures in dimension 4.

Abstract: Semi-toric systems are a special class of autonomous completely integrable Hamiltonian systems defined on a 4-dimensional symplectic manifold. They have two first integrals with commuting flows: one that induces a circular action and another one that does not. Furthermore, only non-degenerate and non-hyperbolic singularities are allowed. These systems appear often in theoretical physics and their richness of possible dynamical behaviours is much greater than, say, toric systems, since new types of non-degenerate singularities can arise, such as focus-focus points.
Semi-toric systems have been classified a few years ago by Pelayo and Vu Ngoc in terms of five symplectic invariants from which the whole system can be reconstructed. However, their explicit calculation is often not straight-forward and until now some invariants had not been yet calculated even for the most basic cases. In this talk we will present the classification and illustrate it with our last results, namely the explicit calculation of all symplectic invariants for two physically-inspired examples: the coupled spin-oscillator and the coupled angular momenta. This is a joint work with H. Dullin and S. Hohloch.

Abstract: Since the geodesic flow of a Riemannian/Finsler manifold may be seen as a Reeb flow on the unit (co)tangent bundle of the underlying manifold, one recovers the systolic ratio as the ratio  of a suitable power of the minimal period of the Reeb flow and the contact volume. With this motivation, one may study the behavior of the systolic ratio as the contact form changes within the class of contact forms that defines a given contact manifold. Recently, Abbondandolo et al. showed that on any contact 3-manifold there exists a contact form with arbitrarily large systolic ratio. Following their work,  we show that the statement holds in any dimension.  Given any contact manifold (M,\xi), using a supported open book decomposition, we first construct a contact form such that on an arbitrarily large portion of M, the Reeb flow is of Boothby-Wang type, while on the complement of this portion, the minimal period of Reeb orbits is bounded below. Second, using certain hamiltonian ball maps, we construct contact mapping tori, for which the minimal period  is bounded below while the contact volume is arbitrarily small. Finally, we replace a collection of trivial mapping tori, which exists due to the Boothby-Wang fibration and covers the most volume of M, with the 'plugs' that are constructed via hamiltonian ball maps. It turns out that the minimal period and the contact volume of a plug are determined by the action and the Calabi invariant of the underlying hamiltonian ball map.  In order to show that the resulting contact form supports \xi, we utilize Giroux's work on supported open books in contact manifolds of higher dimensions.