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

职业：奇异黎曼叶状结构及其在曲率和不变理论中的应用

基本信息

批准号：
2042303
负责人：
Marco Radeschi
金额：
$ 47.53万
依托单位：
University of Notre Dame
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2042303&HistoricalAwards=false
关键词：
CAREER Singular Riemannian Foliations Applications

项目摘要

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.

黎曼几何通过查看可测量的属性（例如长度、距离、角度、体积和曲率）来研究平滑空间（称为黎曼流形）的形状，这些属性量化了与熟悉的平坦空间相比空间变形的程度。黎曼流形在物理学中随处可见，对细胞膜以及广义相对论中的时空进行建模，黎曼几何提供了研究其性质的基本工具。在研究物理对象以及一般黎曼流形时，一个最重要的概念是对称性，即几何对象的“自相似性”程度，这使得它在某些保长变换（称为等距）下保持不变。对称性可以进一步推广为将几何对象划分为“片”的想法，这些“片”彼此保持平行。这个想法通过奇异黎曼叶状结构的数学概念形式化，是该项目数学研究的核心。在这里，我们利用 PI 和合作者引入的想法来研究这些结构的局部行为，并在全局范围内使用它们来产生具有所需曲率的新流形。在这个项目中，几何也被用作一个广泛的概念，涵盖了数学界和整个社会的许多活动，例如：1）与本科生和研究生学校一起组织为期四个星期的度量几何主题项目，以及为期一周的会议。 2）每周为三年级及以上女生组织一次数学营，旨在解决数学学科中的性别失衡问题。该项目的主要目标是研究奇异黎曼叶状结构，以进一步了解其结构并将其应用于不变量理论和黎曼几何。对奇异黎曼叶状结构（即以一片叶子为原点的欧几里得空间上的叶状结构）的局部研究概括了李群的正交表示。 PI 在最近的一项联合工作中证明，无穷小子元有一个代数对应项，由某些称为拉普拉斯代数的多项式代数给出。这为一种称为“无群不变理论”的新方法打开了大门，该方法包括理解如何从拉普拉斯代数中读取流形子量的几何性质，并将这些技术应用于子量来自正交表示的特殊情况。在全球范围内，奇异黎曼叶状结构可以推测是由具有均匀下截面曲率界的流形折叠序列产生的：一个项目将尝试在局部证明这一点。此外，奇异黎曼叶状结构将用于产生具有非负曲率和正曲率的流形的新示例。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。