Foliations, Invariant Theory, and Submanifolds

叶状结构、不变理论和子流形

基本信息

批准号：
2005373
负责人：
Ricardo Augusto Mendes
金额：
$ 17.64万
依托单位：
University of Oklahoma Norman Campus
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-15 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005373&HistoricalAwards=false
关键词：
Foliations Invariant Theory Submanifolds

项目摘要

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.

差异几何形状是数学的一个分支，研究了称为歧管的任意维空间。抽象数学的标志是，概念的一般性使它们可以应用于许多显然不同的情况。歧管可以描述一个物理对象，例如小行星的二维表面或机器人臂的所有配置的空间。远离物理对象，任何数据集都可以看作是一组有限的点集，在这种情况下，尺寸等于人口中测量的数量数量，例如身高，体重，年龄等。该项目的重点是对歧管的“对称性”的研究，可以是有限的，例如蝴蝶或海星或无限的歧管，例如像行星一样的圆形物体的旋转对称性。对称性导致点之间的等效概念（例如，海星的五个尖端是等效的），这自然会导致该歧管的分解或“叶状”，分为称为“叶子”的子序列，这是相互等效的点的集合。对称性还产生“不变函数”的概念，这意味着叶子上的函数恒定。该项目的主要目的是研究不变函数的代数研究与“叶子”的几何研究之间的相互作用。 PI将继续向高中生，本科研究，研究生培训，扩大参与活动以及组织和研讨会的组织。在更多技术术语中，该项目将探索奇异的riemannian殖民地的新兴领域之间的相互作用，以及较旧的Invariant理论和Submanifold理论和Submanifold理论（尤其是Isoparamet submanric submanric subsmanric）。叶叶理论对不变理论的提议的应用包括提供新的，“无群”的经典结果证明，从而为他们提供了新的观点。并证明全新的结果，例如与反向不变理论问题有关。 Submanifold几何形状的拟议应用包括对最小亚曼叶夫的指数的研究，尤其是其与亚曼叶拓扑的关系，这是由Marques-neves-Schoen猜想的例证；以及一种在百年历史的等层属submanifolds分类的问题中，对最后剩下的案例进行了一种新的攻击方法。该项目由几何分析和启发竞争性研究的既定计划共同资助。本奖反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估的评估来通过评估来支持的。