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Ricci Curvature and Torsion

里奇曲率和挠率

基本信息

批准号：
2203536
负责人：
Jeffrey Streets
金额：
$ 20.64万
依托单位：
University of California-Irvine
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-15 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203536&HistoricalAwards=false
关键词：
Ricci Curvature Torsion

项目摘要

Geometry plays a fundamental role in different areas of mathematics, physics, and other scientific disciplines. A central and deeply studied concept in our modern approach to geometry is that of `curvature.’ Uncovering the modern definition of curvature played a key role in the development of Einstein’s theory of relativity over a century ago. On the other hand, the idea of `torsion,’ is a less understood, though no less important, part of geometry. This project will support fundamental research into geometric structures which naturally incorporate torsion, aiming at new applications in mathematics and physics. In addition to supporting research this award will support various training initiatives for undergraduates and graduate students. This includes mentorship of undergraduates to prepare them for graduate study, and professional training and guidance for graduate students. As UC Irvine is nationally recognized as having a diverse student body, this training will support goals of inclusive excellence across at all higher educational levels.The Ricci curvature, Einstein metrics, and Ricci flow play a central, essential, role in our understanding of geometry and analysis, deeply influencing our understanding of a diverse array of subjects such as heat kernels, Brownian motion, partial differential equations, low-dimensional topology, complex geometry, algebraic geometry, and more. Taking inspiration from a variety of new developments in topology, Riemannian geometry, mathematical physics, and complex geometry, a natural extension of Ricci curvature has emerged which incorporates torsion. Attendant to this are natural generalizations of Einstein metrics and Ricci flow. The research project aims at uncovering fundamental geometric and analytic aspects of the generalized Ricci curvature, generalized Einstein metrics, and generalized Ricci flow, ultimately building towards uniformization and classification results with further applications to differential geometry, PDE, complex manifolds, and mathematical physics. The award will aid the training of students at the undergraduate and graduate levels, through outreach activities, by incorporating them in research projects, and through professional training programs.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.

几何在数学、物理和其他科学学科的不同领域中起着基础性的作用。在我们的现代几何方法中，一个中心的和深入研究的概念是“曲率”。揭示曲率的现代定义在世纪前爱因斯坦相对论的发展中发挥了关键作用。另一方面，“挠”的概念是几何学中一个不太清楚的部分，尽管它同样重要。该项目将支持自然包含扭转的几何结构的基础研究，旨在数学和物理学中的新应用。除了支持研究，该奖项将支持本科生和研究生的各种培训计划。这包括指导本科生为研究生学习做准备，以及为研究生提供专业培训和指导。由于加州大学欧文分校是全国公认的具有多元化的学生群体，这项培训将支持所有高等教育水平的包容性卓越目标。里奇曲率，爱因斯坦度量和里奇流在我们对几何和分析的理解中发挥着核心，必不可少的作用，深刻影响了我们对各种学科的理解，如热核，布朗运动，偏微分方程，低维拓扑、复几何、代数几何等等。从拓扑学、黎曼几何、数学物理和复几何的各种新发展中获得灵感，出现了包含挠率的里奇曲率的自然延伸。随之而来的是爱因斯坦度量和里奇流的自然推广。该研究项目旨在揭示广义里奇曲率，广义爱因斯坦度量和广义里奇流的基本几何和分析方面，最终建立均匀化和分类结果，并进一步应用于微分几何，PDE，复流形和数学物理。该奖项将通过推广活动、将学生纳入研究项目以及专业培训计划来帮助本科生和研究生的培训。该奖项反映了NSF的法定使命，并被认为值得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来支持。