Foliations, Invariant Theory, and Submanifolds

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Differential Geometry is a branch of Mathematics that studies spaces of arbitrary dimension called manifolds. A hallmark of abstract mathematics is that the generality of concepts allows them to be applied to many apparently diverse situations. A manifold can describe a physical object, like the two-dimensional surface of an asteroid, or the space of all configurations of a robotic arm. Moving away from physical objects, any data set can be seen as a finite set of points in a manifold, in which case the dimension equals the number of quantities measured, for example height, weight, age, etc in a population. This project focuses on the study of "symmetry" of manifolds, which can be finite, like the one exhibited by a butterfly or a starfish, or infinite, such as the rotational symmetry of a round object like a planet. Symmetry leads to a notion of equivalence between points (for example the five tips of a starfish are equivalent), which naturally gives rise to a decomposition, or "Foliation", of the manifold into sub-manifolds called "leaves", which are sets of points equivalent to each other. Symmetry also yields the notion of "invariant functions", meaning functions constant on the leaves. The main goal of this project is to study the interplay between the algebraic study of invariant functions, and the geometric study of the "leaves". The PI will continue outreach to high school students, undergraduate research, graduate training, broadening participation activities, and organization of conferences and workshops.In more technical terms, this project will explore the interplay between the emerging field of singular Riemannian foliations, and the older fields of Invariant Theory and Submanifold Theory (especially isoparametric and minimal submanifolds). Proposed applications of Foliation Theory to Invariant Theory include providing new, "group-free" proofs of classical results, thus giving them a new perspective; and proving brand-new results, related for example to the Inverse Invariant Theory Problem. Proposed applications to submanifold geometry include the study of the index of minimal submanifolds, especially its relationship to the topology of the submanifold, as exemplified by the Marques-Neves-Schoen conjecture; and a new method of attack for the last remaining case in the century-old problem of classification of isoparametric submanifolds of spheres. This project is jointly funded by the Geometric Analysis and the Established Program to Stimulate Competitive Research (EPSCoR).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.