Ricci Curvature and Torsion

几何在数学、物理和其他科学学科的不同领域中发挥着基础性的作用。在我们现代的几何方法中，一个核心的、被深入研究的概念是“曲率”。揭示曲率的现代定义在一个多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，“扭转”的概念是几何学中一个不太为人所知，但同样重要的部分。该项目将支持几何结构的基础研究，几何结构自然包括扭转，旨在数学和物理中的新应用。除了支持研究外，该奖项还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，以及对研究生进行专业培训和指导。由于加州大学欧文分校被全国公认为拥有多元化的学生群体，此次培训将支持所有高等教育水平的包容性优秀目标。Ricci曲率、爱因斯坦度量和Ricci流在我们对几何和分析的理解中发挥着核心的、基本的作用，深刻影响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。从拓扑学、黎曼几何、数学物理和复杂几何的各种新发展中汲取灵感，出现了包含扭转的Ricci曲率的自然延伸。随之而来的是爱因斯坦度量学和利玛奇流的自然概括。该研究项目旨在揭示广义Ricci曲率、广义爱因斯坦度量和广义Ricci流的基本几何和分析方面，最终建立统一和分类结果，并进一步应用于微分几何、偏微分方程组、复流形和数学物理。该奖项将通过外展活动，通过将学生纳入研究项目和专业培训计划，帮助培养本科生和研究生。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.