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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向我们介绍了三大古典几何之一的欧几里得几何。欧几里得几何可应用于计算机科学和晶体学以及现代数学的各个分支。另外两种几何形状是球形和双曲形。球面几何是地球物理学和天文学研究的核心，对于导航也至关重要。双曲几何在物理学中的狭义相对论中具有现代应用。全局黎曼几何概括了这三种几何。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”）与全局拓扑不变量（例如基本群）之间的关系，该全局拓扑不变量表明空间是否具有一维“孔”。自全局黎曼几何概念提出以来，具有曲率界限的流形已得到深入研究。研究具有下曲率界的流形的一种相对较新的方法是引入对称性，这是该项目的主要焦点。该项目还将继续 PI 对中学生和高中学生的外展工作，以及研究生培训，并组织讲习班和会议，重点是纳入女性和代表性不足的群体。该项目将开展一项计划，其中她仔细研究和分析具有下曲率界的黎曼流形的对称性，考虑截面曲率、里奇曲率、标量曲率和中间标量曲率下限边界及其对亚历山德罗夫空间的一些相应概括，着眼于更深入地了解这一很大程度上未知的空间类别。该项目不仅将研究连续和离散对称性与此类空间拓扑的关系，而且旨在使用对称性和拓扑作为工具，找到正里奇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价值和更广泛的影响审查标准进行评估，被认为值得支持。