CAREER: Discrete and Generalized Riemannian Geometry and Curvature Flows

职业：离散和广义黎曼几何和曲率流

• 批准号: 0748283
• 负责人: David Glickenstein
• 金额: $ 40.17万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-05-01 至 2016-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0748283&HistoricalAwards=false
• 关键词: CAREER; Discrete; Generalized; Riemannian; Geometry

In this project the PI proposes to study discrete geometry, curvature flows, and collapsing solutions to smooth geometric flows. The original motivation for this work is the landmark work on Ricci flow that began with R. Hamilton and includes the solution of the Poincare conjecture by G. Perelman. The PI proposes to work on both combinatorial curvature flows on piecewise linear manifolds and discrete approximations of Ricci flow and other smooth flows. In particular, the PI plans to study discrete flows numerically, to develop visualization techniques for abstract manifolds, to further develop the theory of discrete geometries in the spirit of differential geometry, and to prove convergence of these geometries and related geometric operators to the continuum. The PI also plans to study geometric flows on generalizations of Riemannian manifolds, such as Riemannian groupoids, in order to better understand those flows at singularities. The experimental part of this proposal will be run by a laboratory of undergraduates supervised by graduate students. The recent solution of the Poincare conjecture by G. Perelman both stunned and invigorated the mathematics community. The PI proposes to study similar techniques involving geometric flows in two settings: (1) Discrete Geometries, which may be applied both to other types of geometric questions and to mathematical modelling in a variety of settings, including physics and computer graphics, and (2) Generalized Geometries, which may clarify the implications of Perelman's results and how it may be applied to both mathematical and physical applications. The hope is not only to solve geometric problems, but develop techniques applicable to other areas of science and engineering, both theoretically and computationally. In the process, the PI plans to rely on the laboratory science model to form a group of graduate and undergraduate students developing research tools and presentation tools. The PI hopes to use these tools to communicate the excitement of modern geometry to researchers, teachers, students, and the general public.

在这个项目中，PI建议研究离散几何，曲率流和平滑几何流的折叠解决方案。这项工作的最初动机是从R。汉密尔顿和包括解决庞加莱猜想G.佩雷尔曼PI建议在分段线性流形上的组合曲率流和Ricci流和其他光滑流的离散近似上工作。特别是，PI计划研究离散流数值，开发抽象流形的可视化技术，进一步发展微分几何精神的离散几何理论，并证明这些几何和相关几何算子收敛到连续体。PI还计划研究黎曼流形的推广上的几何流，例如黎曼群胚，以便更好地理解奇点处的几何流。这项建议的实验部分将由一个由研究生监督的本科生实验室管理。 本文给出了G.佩雷尔曼既震惊又震惊了数学界。PI建议在两种设置中研究涉及几何流的类似技术：（1）离散几何，可以应用于其他类型的几何问题和各种设置中的数学建模，包括物理和计算机图形，以及（2）广义几何，可以澄清佩雷尔曼结果的含义以及如何将其应用于数学和物理应用。希望不仅是解决几何问题，但开发适用于其他领域的科学和工程技术，无论是理论上和计算。在这个过程中，PI计划依靠实验室科学模型，形成一组研究生和本科生开发研究工具和演示工具。PI希望利用这些工具向研究人员、教师、学生和公众传达现代几何的兴奋。