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Title: The Sard conjecture for the endpoint map
Researcher: Álvaro del Pino
Consider a manifold M, which we think of as the space of potential positions for a particle. In many physically meaningful situations, this particle may not be able to move freely, and instead have only a subset of all directions available for its motion. We model this as a subbundle (not necessarily vectorial) of the tangent bundle and we call it a control system .
Given a control system, we ask ourselves whether a particle can move from a point A to a point B, despite the constraints. If the answer is yes, we can then try to minimise the cost of travelling between the two points. Of course, for this to even make sense we need to introduce a notion of cost for our trajectories. This is often done by introducing a Riemannian metric. Subriemannian Geometry (also known as Geometric Control Theory), is the area dedicated to studying control systems endowed with such a metric.
Consider now the space of all trajectories that satisfy our constraints. This is some sort of infinite dimensional object, and we would like to claim that it is a manifold (which is certainly true for the space of paths with no constraints). It turns out that this is almost true: for reasonable control systems, it is a manifold almost everywhere, but it may have some singularities. These correspond to so-called singular curves: i.e. trajectories such that if we freeze one of the endpoints and infinitesimally move the other, we may be unable to infinitesimally move the curve to follow it. This is bad in Control Theory: It means that if we want to change our destination a bit, there is no small deformation of our path to get there!
The most salient conjecture in Subriemannian Geometry asks whether the set of points B reachable from a given A using singular curves has measure zero. This is like the usual Sard property, but now one of the spaces involved (the space of trajectories satisfying the constraints) is infinite dimensional. The student will study the existing literature on the problem, starting from the basics on Subriemannian Geometry, and building up towards recent papers solving the conjecture for concrete Lie groups.
Depending on the interest of the student, we will delve deeper into different aspects. In the context of Lie groups the conjecture reduces to the study of dynamical systems related to the Lie algebra structure. In the general context, results can be obtained using the symplectic geometry naturally present in the study of control systems.
A.A. Agrachev, Y. Sachkov. "Control Theory from the Geometric Viewpoint". Encyclopaedia of Mathematical Sciences, 87. Springer-Verlag Berlin Heidelberg (2004).
R. Montgomery. "A tour of subriemannian Geometries, their geodesics and applications". Mathematical Surveys and Monographs, 91 (2002).
A. Belotto da Silva, A. Figalli, A. Parusinski, L. Rifford. "Strong Sard Conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3".