LEAPS-MPS: Geometrization with Positive Curvature

几何的一个基本主题是研究曲率和空间形状之间的关系。该项目将在四维及更高维度中研究“正曲率”如何部分或全部决定底层空间的整体形状，旨在将著名的二维高斯-邦尼特定理以及汉密尔顿和佩雷尔曼在三维几何化方面的工作扩展到更高维度。在这个项目中，PI将利用曲率算子和里奇流等工具，将几何化的研究扩展到四维及更高维度的流形。此外，PI将为威奇托当地的中学生和高中生组织各种活动，并参与威奇托州立大学本科生和研究生的指导计划，旨在扩大参与，特别是在代表性不足的少数民族学生中。本课题的研究目的是研究四维及高维空间的分类，包括紧致黎曼流形、爱因斯坦流形和收缩梯度Ricci孤子，其曲率满足正各向同性曲率、正截面曲率、正Ricci曲率和第二类正曲率算子等正条件。其结果是更好地理解曲率和拓扑之间的关系。主要策略包括分析里奇流的解，将极大值原理应用于几何量满足的偏微分方程，以及通过张量代数和李代数理解各种正曲率概念之间的关系。该项目由启动早期职业生涯学术路径在数学和物理科学（LEAPS）和建立计划，以刺激竞争研究（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，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。

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.