Curvature and Symmetry

我们在高中学习的平面几何为我们介绍了三大经典几何之一的欧几里德几何。欧几里德几何在计算机科学和晶体学以及现代数学的各个分支中都有应用。其他两种几何形状是球面和双曲线。球面几何是天体物理学和天文学研究的中心，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。整体黎曼几何推广了这三种几何。该研究领域的主要挑战之一是了解局部几何不变量（例如曲率，即所考虑的空间如何“弯曲”）如何与全局拓扑不变量（例如基本群）相关，这表明该空间是否有1维“洞”。自整体黎曼几何提出以来，具有曲率界的流形一直是人们研究的热点。一个相对较新的方法来研究流形与较低的曲率界限一直是引进对称性，是这个项目的主要焦点。该项目还将继续PI的初中和高中学生的推广工作，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性不足的群体。该项目将继续一个程序，她仔细研究和分析黎曼流形的对称性与曲率下限，考虑截面，Ricci，标量，和中间标量曲率下界和一些相应的推广到亚历山德罗夫空间，着眼于获得更深入的了解这在很大程度上未知的一类空间。该项目不仅将研究连续和离散对称性如何与此类空间的拓扑结构相关，而且还旨在使用对称性和拓扑学作为工具来寻找具有正Ricci曲率和几乎非负截面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会议和研讨会的组织，强调包容性。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.