Curvature and Symmetry

我们在高中学的平面几何给我们介绍了欧几里得几何，它是三大经典几何之一。欧几里德几何在计算机科学和结晶学以及现代数学的各个分支中都有应用。另外两个几何形状是球面和双曲线。球面几何是地球物理和天文学研究的核心，对航海也至关重要。双曲几何在物理学中的狭义相对论中有现代应用。全局黎曼几何推广了这三个几何。这一研究领域的主要挑战之一是了解局部几何不变量(如曲率)如何与全局拓扑不变量(如基本群)有关，如曲率，即所考虑的空间如何“弯曲”，基本群指示空间是否有一维“洞”。自从整体黎曼几何的概念提出以来，具有曲率界的流形得到了广泛的研究。一种相对较新的方法来研究具有下曲率界限的流形是引入对称性，这也是本项目的主要焦点。该项目还将继续PI与初中生和研究生的外展工作，以及研究生培训，并组织研讨会和会议，重点是纳入妇女和代表不足的群体。该项目将继续执行一个项目，在该项目中，她将仔细研究和分析具有较低曲率界的黎曼流形的对称性，考虑分段、Ricci、标量和中间标量曲率下界以及它们对Alexandrov空间的一些相应推广，以期更深入地了解这类基本上未知的空间。该项目不仅将研究连续和离散对称如何与这类空间的拓扑相关，而且还旨在利用对称和拓扑作为工具来寻找具有正Ricci曲率和几乎非负截曲率的黎曼流形的新例子。该项目还包括对学生的培训和指导，以及强调包容性的会议和研讨会组织。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。

The plane geometry we learn in high school gives us an introduction to Euclidean geometry, one of the three classical geometries. Euclidean geometry has applications to computer science and crystallography, as well as various branches of modern mathematics. The other two geometries are Spherical and Hyperbolic. Spherical geometry is central to the study of geophysics and astronomy, and vital for navigation. Hyperbolic geometry has modern applications to the theory of special relativity in Physics. Global Riemannian Geometry generalizes these three geometries. One of the major challenges in this area of study is to understand how local geometric invariants such as curvature, that is, how the space under consideration "bends", relate to global topological invariants such as fundamental group, which indicates whether or not the space has 1-dimensional "holes". Manifolds with curvature bounds have been studied intensively since the conception of global Riemannian geometry. One relatively recent approach to the study of manifolds with lower curvature bounds has been the introduction of symmetries and is the main focus of this project. The project will also continue the PI's outreach work with middle and high school students, as well as graduate training, and the organization of workshops and conferences with an emphasis on the inclusion of women and under-represented groups.The project will pursue a program in which she carefully studies and analyzes symmetries of Riemannian manifolds with lower curvature bounds, considering sectional, Ricci, scalar, and intermediate scalar curvature lower bounds and some of their corresponding generalizations to Alexandrov spaces, with an eye to gaining a deeper understanding of this largely unknown class of spaces. The project will study not only how continuous and discrete symmetries relate to the topology of such spaces, but also aim to find new examples of Riemannian manifolds of positive Ricci curvature and almost non-negative sectional curvature using symmetries and topology as tools to do so. The project also includes training and mentoring of students as well as conference and workshop organization with an emphasis on inclusivity.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.