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.

Riemannian几何形状通过查看可测量的特性，例如长度，距离，角度，体积和曲率，研究了称为Riemannian歧管的光滑空间的形状 - 与熟悉的平面空间相比，它量化了多少空间变形。 Riemannian歧管在物理学中无处不在，对细胞的膜以及一般相对论的时空进行建模，Riemannian几何形状提供了研究其特性的基本工具。当研究物理对象以及riemannian流形的一个概念通常是对称性的概念，即几何对象的“自我相似性”的程度，这使其在某些长度可实现的变换（称为isometries）下使其不变。可以通过将几何对象划分为“床单”的想法进一步概括对称性，从而保持彼此平行。这个想法是由奇异riemannian叶片的数学概念正式形式化的，是该项目数学研究的中心。在这里，我们使用PI和合作者引入的想法来研究这些结构的当地行为，并在全球使用它们来生产具有理想曲率的新歧管。在该项目中，几何学也被用作广泛的概念，以涵盖数学社区和整个社会的许多活动，例如：1）在公制的几何学中组织一个为期四个星期的主题计划，并与本科生和研究生的学校以及为期一周的会议。 2）组织一个每周的三年级女孩数学训练营，旨在解决数学学科中的性别失衡。该项目的主要目标是研究奇异的黎曼叶子，既要进一步了解他们的结构，又要将其应用于不变理论和Riemannian deemetry。奇异riemannian叶子的本地研究，即以一片叶子起源的欧几里得空间上的叶子，概括了谎言群的正交表示。 PI在最近的联合工作中证明了无穷小的地下具有代数对应物，该代数由某些称为laplacian代数的多项式代数给出。这打开了一种称为“无群体的不变理论”的新方法，包括理解如何读取其拉普拉斯代数的流形子顺序的几何特性，并将这些技术应用于从正交表示的特殊情况下。在全球范围内，奇异的riemannian叶子可以猜想是由于较低的截面曲率结合的歧管的崩溃序列：一个项目将试图在本地证明这一点。此外，奇异的里曼尼亚叶子将用于制作具有非负和积极曲率的歧管的新例子。该奖项反映了NSF的法定任务，并认为值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来获得支持。