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

Some classical and fundamental results in analysis and geometry fail to hold in spaces of infinite dimension, leading to striking phenomena that require new investigations. We address two distinct questions of this type. To each of them is dedicated a part of the thesis: the first part contains the main work of my Ph.D. studies, while the second one is a complementary work that I did during the same years. Here we give a brief summary, in order to give a flavour of the topics. Each part contains a detailed introduction. Part I - Sard properties for polynomial maps in infinite dimension and applications to sub-Riemannian geometry. It is well-known that the classical Morse-Sard theorem is false for smooth maps from an infinite dimensional Hilbert space to R, even under the assumption that the map is “polynomial”, and a general theory is still missing. In this part we address this issue, providing sharp quantitative criteria for the validity of Sard-type theorems for polynomial maps from an infinite dimensional Hilbert space to R^m. As an application, we present new advances on the sub-Riemannian Sard conjecture in Carnot groups. The research presented in this part appears in the following preprints: • A. Lerario, L. Rizzi, D. Tiberio, Sard properties for polynomial maps in infinite dimension, arxiv:2407.02296 • A. Lerario, L. Rizzi, D. Tiberio, Quantitative approximate definable choices, arxiv:2409.14869 Part II - Vanishing geodesic distances and the Michor-Mumford conjecture in Hilbertian H-type groups. It is well-known that for weak Riemannian metrics on infinite dimensional manifolds the geodesic distance may not be a genuine distance, indeed it can be zero on distinct points. In their 2005 paper, Michor and Mumford conjectured that the degeneracy of the geodesic distance is related to the local unboundedness of the sectional curvature. In this part of the thesis, we introduce Heisenberg-type Lie groups modelled on Hilbert spaces, and we show that in this setting the degeneracy of the geodesic distance and the local unboundedness of the sectional curvature coexist for a large class of weak Riemannian metrics. The research presented in this part appears in the following preprints and publications: • V. Magnani, D. Tiberio, On the Michor-Mumford phenomenon in the infinite dimensional Heisenberg group, Revista Matematica Complutense • V. Magnani, D. Tiberio, The Michor-Mumford conjecture in Hilbertian H-type groups, arxiv:2404.04583