LEAPS-MPS: Geometrization with Positive Curvature

LEAPS-MPS：具有正曲率的几何化

• 批准号: 2316659
• 负责人: Xiaolong Li
• 金额: $ 25万
• 依托单位: Wichita State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316659&HistoricalAwards=false
• 关键词: LEAPS; MPS; Geometrization; Positive; Curvature

A fundamental theme in geometry is to study the relationship between the curvature and the shape of the space. This project will investigate in dimensions four and higher how “positivity curvature” determines, partially or in full, the global shape of the underlying space, aiming to extend the famous Gauss-Bonnet Theorem in dimension two and Hamilton and Perelman’s work on geometrization in dimension three to higher dimensions. In this project, the PI will expand the study of geometrization to manifolds of dimensions four and higher, making use of tools such as curvature operators and Ricci flows. In addition, the PI will organize a variety of activities for middle and high school students in the local Wichita area and engage in mentoring programs for undergraduate and graduate students at Wichita State University aimed at broadening participation, especially amongst underrepresented minority students. The research objective of this project is to investigate the classification of four and higher-dimensional spaces, including compact Riemannian manifolds, Einstein manifolds, and shrinking gradient Ricci solitons, whose curvature satisfies a positivity condition, such as positive isotropic curvature, positive sectional curvature, positive Ricci curvature, and positive curvature operator of the second kind. The outcome is a better understanding of the relationship between curvature and topology. Primary strategies include analyzing solutions to the Ricci flow, applying the maximum principle to partial differential equations satisfied by geometric quantities, and understanding the relationship between various notions of positive curvature via tensor algebra and Lie algebra.This project is jointly funded by the Launching Early-Career Academic Pathways in the Mathematical and Physical Sciences (LEAPS) and the Established Program to Stimulate Competitive Research (EPSCoR).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.

几何学的一个基本主题是研究曲率和空间形状之间的关系。该项目将在“阳性曲率”中的四个及更高的方面进行调查，以部分或完整地确定基础空间的全球形状，旨在将著名的高斯 - 邦网定理扩展在二维维度上，汉密尔顿和佩雷尔曼在维度上的几何作用到三个维度。在这个项目中，PI将使用曲率运算符和Ricci流等工具，将几何学的研究扩展到尺寸四个及更高尺寸的流形。此外，PI将为当地威奇托地区的中学学生和高中生组织各种活动，并为威奇托州立大学的本科生和研究生进行心理计划，旨在扩大参与的参与，尤其是在代表性不足的少数群体中。 The research objective of this project is to Investigate the classification of four and higher-dimensional spaces, including compact Riemannian manifolds, Einstein manifolds, and shrinking gradient Ricci solidons, whose curvature satisfies a positive condition, such as positive isotropic curvature, positive sectional curvature, positive Ricci curvature, and positive curvature operator of the second kind.结果是更好地理解曲率与拓扑之间的关系。主要策略包括分析RICCI流量的解决方案，将最大原则应用于通过几何数量满足的部分微分方程，并了解通过张量代数和谎言代数的积极弯曲的各种音符之间的关系。该项目由该项目共同资助了由刺激和物理学的早期研究（Leagical和物理学）（Leaical ofdersion）（Leeps）（Leeps）（Leeps）（Leeps）（Leagical inthematival of）（Leeps）（Leeps）（Leeps）的（Leeps）（促进）的基金（反映了NSF的法定任务，并通过使用基金会的知识分子和更广泛的影响审查标准评估来诚实地表示支持。