Foliations, Invariant Theory, and Submanifolds

微分几何是数学的一个分支，研究称为流形的任意维空间。抽象数学的一个特点是概念的普遍性使它们可以应用于许多明显不同的情况。流形可以描述一个物理对象，如小行星的二维表面，或机器人手臂的所有配置的空间。远离物理对象，任何数据集都可以被视为流形中的有限点集，在这种情况下，维度等于测量的数量，例如人口中的身高，体重，年龄等。这个项目的重点是研究流形的“对称性”，它可以是有限的，如蝴蝶或海星所展示的，也可以是无限的，如行星等圆形物体的旋转对称性。对称性导致了点之间等价的概念（例如海星的五个尖端是等价的），这自然会引起流形分解或“叶子”，成为称为“叶子”的子流形，这些子流形是彼此等价的点集。对称性也产生了“不变函数”的概念，这意味着函数在叶子上是常数。这个项目的主要目标是研究不变函数的代数研究和“叶子”的几何研究之间的相互作用。PI将继续推广到高中生，本科生研究，研究生培训，扩大参与活动，并组织会议和研讨会。在更专业的术语，这个项目将探讨奇异黎曼叶理的新兴领域之间的相互作用，和不变理论和子流形理论的老领域（特别是等参和极小子流形）。叶理论在不变量理论中的应用包括为经典结果提供新的“无群”证明，从而给它们一个新的视角;以及证明全新的结果，例如与逆不变量理论问题有关的结果。子流形几何学的应用包括研究极小子流形的指数，特别是它与子流形拓扑的关系，例如马克-内维斯-舍恩猜想;以及一种新的攻击方法，用于解决球的等参子流形的分类问题。该项目由几何分析和刺激竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.