CAREER: Discrete and Generalized Riemannian Geometry and Curvature Flows

在这个项目中，PI建议研究离散几何、曲率流和光滑几何流的坍缩解。这项工作的最初动机是关于里奇流的里程碑式的工作，从汉密尔顿开始，包括G.佩雷尔曼对庞加莱猜想的解决。PI既研究分段线性流形上的组合曲率流，也研究Ricci流和其他光滑流的离散近似。特别是，PI计划在数值上研究离散流动，开发抽象流形的可视化技术，以微分几何的精神进一步发展离散几何理论，并证明这些几何和相关的几何算子对连续体的收敛性。PI还计划在黎曼流形的推广上研究几何流，例如黎曼群，以便更好地理解奇点处的那些流。本提案的实验部分将由研究生指导的本科生实验室进行。佩雷尔曼（G. Perelman）最近对庞加莱猜想的解答既震惊又振奋了数学界。PI建议在两种情况下研究涉及几何流的类似技术：(1)离散几何，它可以应用于其他类型的几何问题和各种情况下的数学建模，包括物理和计算机图形学；(2)广义几何，它可以澄清佩雷尔曼结果的含义，以及它如何应用于数学和物理应用。希望不仅是解决几何问题，但发展技术适用于其他领域的科学和工程，理论和计算。在此过程中，PI计划依靠实验室科学模式，组建一个由研究生和本科生组成的小组，开发研究工具和演示工具。PI希望使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之处。

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.