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 and its analytic properties. It may be traced back to ancient Antiquity where scientists like Ptolemy designed maps of their world upon a mathematical description of the Earth. This project focuses on three key aspects of the field. The first one is the study of singular spaces obtained as limits of smooth shapes satisfying some uniform curvature constraints; such limits naturally appear in geometric evolution problems like Kähler-Ricci flow or mean curvature flow. The second one is the construction of geometrically meaningful representations of a smooth shape as a subset of a simple ambient space; this will be investigated through spectral optimization and kernel embeddings. The third aspect is the setting up and implementation of new machine learning algorithms for datasets whose underlying geometry satisfies suitable geometric constraints.