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

黎曼几何研究光滑空间的形状，称为黎曼流形，通过观察可测量的属性，如长度、距离、角度、体积和曲率-这些属性量化了与我们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，在广义相对论中对细胞膜和时空进行建模，而黎曼几何为研究它们的性质提供了基本工具。在研究物理对象和黎曼流形时，最重要的一个概念是对称性，即几何对象的“自相似”程度，它使几何对象在某些保长变换(称为等距变换)下不变。对称性可以通过将几何对象分割成彼此保持平行的“薄片”的想法来进一步推广。这个概念被奇异黎曼叶理的数学概念形式化，是这个项目的数学研究的中心。在这里，我们使用PI和合作者引入的思想来研究这些结构的局部行为，并在全局上使用它们来产生具有理想曲率的新流形。在这个项目中，几何也被用作一个广泛的概念，以涵盖数学界和整个社会的一些活动，例如：1)组织为期四周的度量几何主题课程，有本科生和研究生的学校，以及为期一周的会议。2)为三年级及以上女生组织每周一次的数学夏令营，旨在解决数学学科中的性别不平衡问题。本项目的主要目标是研究奇异黎曼叶，以进一步了解其结构，并将其应用于不变理论和黎曼几何。奇异黎曼叶的局部研究，即欧氏空间上以一片叶为原点的叶，推广了李群的正交表示。PI在最近的一项联合工作中证明了无穷小次度量有一个代数对应，由某些称为拉普拉斯代数的多项式代数给出。这为一种新的方法打开了大门，这种方法被称为“无群不变理论”，包括了解如何从流形的拉普拉斯代数中读出流形次度量的几何性质，并将这些技巧应用于次对称来自于正交表示的特殊情况。在全球范围内，奇异的黎曼叶可以由具有一致的截面曲率下界的流形的折叠序列猜想地产生：一个项目将试图在局部证明这一点。此外，奇异的黎曼叶将被用来产生具有非负和正曲率的流形的新例子。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.