12:00
Sard properties for polynomial maps in infinite dimension
Abstract
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” – and a general theory is still lacking. In this seminar, I will provide sharp quantitative criteria for the validity of Sard’s theorem in this setting, obtained combining a functional analysis 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.