Bernard Bonnard (Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon)
Grégoire Charlot (Université Joseph Fourier, Grenoble)
Room: Amphi Darboux
9:30 Welcome coffee
10:00-11:00 Constantin Vernicos
11:00-12:00 Juan-Carlos Alvarez Paiva
13:45-14:45 Grégoire Charlot
14:45-15:45 Roberta Ghezzi
16:00-17:00 Dario Prandi
Juan-Carlos Alvarez Paiva -- Moment maps and the geometry of curves on Grassmannians and flag manifolds
The study of the Schwartzian derivative and its generalizations is a classical subject that is periodically rediscovered and renewed. In this talk I will present a synthesis that starts from two tautological constructions---among them the moment map for the action of the linear group on the cotangent of Grasmannians and flag manifolds---and neatly gives the whole theory in a form that is particularly suitable for the study of geodesic flows. As an application, I'll present a computation-free proof of a theorem of Foulon stating that Katok perturbations are curvature preserving.
Grégoire Charlot -- Local properties of generic almost-Riemannian and sub-Finslerian manifolds in dimension 2.
We will discuss normal forms, small spheres, local cut and conjugate loci for almost-Riemannian and sub-Finslerian metrics in dimension 2.
Roberta Ghezzi -- Topological invariants for almost-Riemannian surfaces.
In this talk we analyze global properties of almost-Riemannian structures on compact oriented surfaces. We investigate the relation between the topological invariants of the structure and the rank-two vector bundle over the surface. For the general case including tangency points, we provide a classification of oriented almost-Riemannian structures in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian surfaces with tangency points which highlights the role of the underlying vector bundle and explains the connection between geometry and topology of the structure. Finally, we describe how the presence of singular points affects the almost-Riemannian distance and, in particular, its Lipschitz equivalence class.
Dario Prandi -- Intrinsic diffusions in 2D almost-Riemannian structures.
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear on a singular set. Generically, the singular set is an embedded one dimensional manifold and it contains two type of points: Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. In this talk we will focus on the properties of the Laplace-Beltrami operator associated with these structures, and how they are influenced by the singularity. In particular we will discuss recent results, in collaboration with U. Boscain and M. Seri, regarding the propagation of heat through the singularity in the generic case. These results extend the results obtained by Boscain and Laurent, allowing for the presence of tangency points, and are obtained through Agmon estimates and Hardy inequality techniques.
Constantin Vernicos -- Macroscopic spectrum of Nilmanifolds.
We consider Riemanian metrics on nilmanifolds coming from a compact quotient. We will describre the behaviour of the spectrum of the Laplacian on metric balls centred at a given point as their radii goes to infinity. Up to an ad-hoc normalisation the whole spectrum converges to the spectrum of an hypoelliptic operator, arising from a sub-riemannian metric (=the macroscopic spectrum). The normalisation also implies the convergence of the metric balls towards the metric ball of a sub-finslerian metric. Both metrics are Carnot-Caratheodory. The comparison of those metrics allows us to get a kind of macroscopic rigidity.