Résume | I will present some recent results concerning the Sard Conjecture in Sub-Riemannian geometry (SR geometry). SR geometry studies the trajectories in a manifold M which satisfies an extra constraint: they must be almost everywhere tangent to a totally non-holonomic distribution D over M. Some of these trajectories, which are called singular, have pathological behaviours which have no analog in Riemannian geometry. The Sard Conjecture states that the set of points one can reach via singular horizontal paths is "small", that is, it has Lebesgue measure zero.
I will explain how this Conjecture can be interpreted as a geometrical problem concerning the behavior of a characteristic singular foliation in the cotangent bundle. Under the hypothesis of analyticity of M and D, we can study this singular foliation via methods of singularity theory, subanalytic geometry and control measure theory. This is the approach used in our recent results in collaboration with Parusinki, Figalli and Rifford. |