Geometric and analytic properties of metric measure spaces with spectral curvature constraints, with applications to manifold learning

Geometric analysis is the study of the interplay between the
geometric features of a space, like its curvature, its diameter, or the
length of its minimal curves, and its analytic features, such as the
heat propagation or the frequencies of its vibration. This project
focuses on three aspects of geometric analysis. The first one is the
study of a class of singular spaces, called Kato limit spaces, obtained
as limits of smooth Riemannian manifolds satisfying a uniform
spectral constraint on the curvature; such limits appear naturally in
geometric evolution problems like Ricci flow or mean curvature flow.
The second one is the study of spectral optimization problems on a
given smooth manifold; these problems are closely related to the
construction of geometrically meaningful representations of the
manifold as a subset of a simple ambient space. The third aspect
consists in setting up and implementing new machine learning
algorithms for datasets whose underlying geometry satisfies suitable
geometric constraints.