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CAREER: Aspects of Riemannian Geometry and Manifolds with Density

职业：黎曼几何和密度流形的各个方面

基本信息

批准号：
1654034
负责人：
William Wylie
金额：
$ 47.25万
依托单位：
Syracuse University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1654034&HistoricalAwards=false
关键词：
CAREER Aspects Riemannian Geometry Manifolds

项目摘要

Riemannian geometry investigates the generalization of familiar geometric notions such as length, angle, and volume to more abstract, often high-dimensional, spaces called manifolds. Manifolds and their geometric properties are not only central to various branches of mathematics but also are ubiquitous as mathematical models in the sciences. Perhaps the greatest example of this is in general relativity, where one of Einstein's great breakthroughs was the discovery that gravity is modeled by the mathematical notion of curvature. This research project focuses on curvature of Riemannian manifolds. Specific emphasis will be placed on manifolds with density, which can be envisioned as Riemannian manifolds composed of a material with variable, as opposed to uniform, density. Despite the appearance of manifolds with density in various areas of mathematics and applications, including proof of the Poincare conjecture, they are not yet well understood. This project addresses this gap by helping to develop a geometric theory of manifolds with density. The project also uses geometry as a theme for a coherent program of educational activities that include: (1) professional development workshops in teaching and learning for in-service secondary education teachers, which will help support teachers in transition to recently developed K-12 standards; (2) innovation in introductory differential geometry curricula at the undergraduate and graduate level that, by emphasizing applications, will also foster new interaction between mathematics and science researchers; and (3) training of mathematics graduate students in best strategies for engaging students in introductory calculus courses. Ricci curvature for manifolds with density has been an active area of recent research. This includes Ricci solitons, which are both fixed points of the Ricci flow and examples of spaces with constant weighted Ricci curvature, and the study of weighted Ricci curvature bounds. The investigator will continue study of the classification of shrinking Ricci solitons in dimension four and higher. In addition to playing a key role in development of monotonic functionals for the Ricci flow, weighted Ricci curvature bounds also appear in the theory of optimal transport, isoperimetric inequalities, and general relativity and cosmology. Despite the vast amount of research in Ricci curvature of manifolds with density, there was no corresponding theory of weighted sectional curvature until recently; another goal of this project is to further develop the theory of weighted sectional curvature bounds, including investigating applications and connections to other areas of mathematics. Recent collaborative work of the investigator also introduces a new geometric approach to manifolds with density that places a certain torsion free affine connection as the fundamental object of study. This approach not only provides new insight into the weighted Ricci and sectional curvatures but also suggests new natural structures that promise novel results, such as a weighted holonomy group, which will be investigated.

黎曼几何研究了将熟悉的几何概念，如长度，角度和体积推广到更抽象的，通常是高维的，称为流形的空间。流形及其几何性质不仅是数学各个分支的核心，而且作为科学中的数学模型也是无处不在的。也许这方面最好的例子是广义相对论，爱因斯坦的一个伟大突破是发现引力是由数学概念曲率建模的。本研究计画主要研究黎曼流形的曲率。具体重点将放在流形的密度，这可以设想为黎曼流形组成的材料与可变的，而不是统一的，密度。尽管密度流形出现在数学和应用的各个领域，包括庞加莱猜想的证明，但它们还没有得到很好的理解。这个项目通过帮助发展密度流形的几何理论来解决这个问题。该项目还将几何作为一个主题，开展一系列教育活动，包括：（1）为在职中学教师举办教与学专业发展讲习班，这将有助于支持教师向最近制定的K-12标准过渡;（2）创新本科生和研究生阶段的微分几何入门课程，通过强调应用，还将促进数学和科学研究人员之间的新的互动;和（3）培训数学研究生的最佳策略，让学生参与微积分入门课程。密度流形的Ricci曲率是近年来研究的一个活跃领域。这包括里奇孤子，这是两个不动点的里奇流和空间的例子，与恒定加权里奇曲率，并研究加权里奇曲率的界限。研究人员将继续研究四维及更高维的收缩Ricci孤子的分类。除了在Ricci流的单调泛函的发展中发挥关键作用外，加权Ricci曲率界也出现在最优运输理论、等周不等式、广义相对论和宇宙学中。尽管对密度流形的里奇曲率有大量的研究，但直到最近才有相应的加权截面曲率理论;该项目的另一个目标是进一步发展加权截面曲率边界理论，包括研究应用和与其他数学领域的联系。最近的合作工作的调查员还介绍了一种新的几何方法流形密度的地方一定的挠自由仿射连接的基本对象的研究。这种方法不仅提供了新的见解加权里奇和截面曲率，但也提出了新的自然结构，承诺新的结果，如加权holonomy组，这将是调查。