CAREER: Singular Riemannian Foliations and Applications to Curvature and Invariant Theory

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Riemannian Geometry studies the shape of smooth spaces, called Riemannian manifolds, by looking at measurable properties such as lengths, distances, angles, volumes and curvature - which quantify how much space is deformed compared to the familiar flat space. Riemannian manifolds appear everywhere in physics, modeling the membrane of a cell as well as spacetime in general relativity, and Riemannian geometry offers fundamental tools to study their properties. One concept of paramount importance, when studying physical objects as well as Riemannian manifolds in general, is that of symmetry, that is, the degree of "self similarity" of a geometric object, which make it invariant under certain length-preserving transformations (called isometries). Symmetry can be further generalized with the idea of partitioning a geometric object into "sheets", which stay parallel to one another. This idea, formalized by the mathematical concept of singular Riemannian foliation, is at the center of the mathematical investigation of this project. Here, we use ideas introduced by the PI and collaborators to study the local behavior of these structures, as well as use them globally to produce new manifolds with desirable curvature. In this project, geometry is also used as a broad concept to encompass a number of activities for the mathematical community and society in general, such as: 1) Organizing a four-weeks-long thematic program in Metric Geometry, with schools for undergraduate and graduate students, as well as a week-long conference. 2) Organizing a weekly math camp for girls in 3rd grade and up which is aimed at addressing the gender imbalance in the mathematical disciplines. The main goal of this project is to study singular Riemannian foliations, both to further understand their structure and to apply them to Invariant Theory and Riemannian Geometry. The local study of singular Riemannian foliations, namely foliations on a Euclidean space with the origin as one leaf, generalizes orthogonal representations of Lie groups. The PI proved in a recent joint work that infinitesimal submetries have an algebraic counterpart, given by certain polynomial algebras called Laplacian algebras. This opens the door to a novel approach called "Invariant theory without groups", consisting in understanding how to read geometric properties of manifold submetries off of their Laplacian algebra, and apply these techniques to the special case in which the submetry comes from an orthogonal representation. Globally, singular Riemannian foliations can conjecturally arise from collapsing sequences of manifolds with a uniform lower sectional curvature bound: One project will try to prove this locally. Furthermore, singular Riemannian foliations will be used to produce new examples of manifolds with non-negative and positive curvature. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.