Curvature and Symmetry

曲率和对称性

基本信息

批准号：
2204324
负责人：
Catherine Searle
金额：
$ 26.18万
依托单位：
Wichita State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-01 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204324&HistoricalAwards=false
关键词：
Curvature Symmetry

项目摘要

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.

我们在高中学习的平面几何形状使我们介绍了欧几里得的几何形状，这是三种经典的几何形状之一。欧几里得的几何形状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。其他两个几何形状是球形和双曲线。球形几何形状对于研究地球物理和天文学的研究至关重要，对于导航至关重要。双曲线几何形状对物理学特殊相对论具有现代应用。全球Riemannian几何形状概括了这三个几何形状。该研究领域的主要挑战之一是了解诸如曲率之类的局部几何不变剂，即所考虑的“弯曲”空间如何与全球拓扑不变性（例如基本群体）相关，这表明该空间是否具有一维“孔”。自全球riemannian几何形状概念以来，已经对具有曲率边界的流形进行了深入的研究。对曲率范围较低的流形研究的一种相对较新的方法是引入对称性，并且是该项目的主要重点。该项目还将继续与中学生和研究生培训以及组织和会议的组织以及强调妇女和不足的群体的组织组织和会议的组织。界限及其对亚历山德罗夫空间的一些相应概括，以期深入了解这一很大未知类的空间。该项目不仅将研究与此类空间拓扑的连续和离散对称性如何相关，而且还旨在寻找利用曲率曲率的Riemannian流形的新示例，以及使用对称性和拓扑结构和拓扑的几乎非负分段曲率来做到这一点。该项目还包括对学生的培训和指导，以及会议和研讨会组织，重点是包容性。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的审查标准，被认为值得通过评估来获得支持。