Sard properties for polynomial maps in infinite dimension and applications to the sub-Riemannian Sard conjecture

Sard’s theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. However, when the domain is infinite-dimensional and the range is finite-dimensional, the result is not true - even under the assumption that the map is “polynomial”. In this seminar, I will provide sharp quantitative criteria for the validity of Sard’s theorem in this setting, obtained by combining a functional-analytic approach with new tools in semialgebraic geometry. As an application, I will present new results on the Sard conjecture in sub-Riemannian geometry. Based on a joint work with A. Lerario and L. Rizzi.