CAREER: Aspects of Riemannian Geometry and Manifolds with Density

黎曼几何研究将常见的几何概念，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，也是科学中普遍存在的数学模型。也许最好的例子是广义相对论，在广义相对论中，爱因斯坦最大的突破之一是发现了引力是由曲率的数学概念建模的。本文主要研究黎曼流形的曲率问题。具体的重点将放在具有密度的流形上，它可以被想象为由具有可变密度的材料组成的黎曼流形，而不是均匀的密度。尽管具有密度的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，但它们还没有被很好地理解。这个项目通过帮助发展具有密度的流形的几何理论来解决这一差距。该项目还将几何作为一项连贯的教育活动计划的主题，其中包括：(1)为在职中学教师举办教与学的专业发展讲习班，这将有助于支持过渡到最近制定的K-12标准的教师；(2)本科生和研究生一级的微分几何入门课程的创新，通过强调应用，也将促进数学和科学研究人员之间的新的互动；以及(3)对数学研究生进行培训，使他们了解让学生参与微积分入门课程的最佳策略。具有密度的流形的Ricci曲率是最近研究的一个活跃领域。这包括Ricci孤子，它既是Ricci流的不动点，也是具有常权Ricci曲率的空间的例子，以及对加权Ricci曲率界的研究。研究人员将继续研究四维及更高维度的收缩利玛窦孤子的分类。除了在Ricci流的单调泛函的发展中发挥关键作用外，加权Ricci曲率界还出现在最优输运理论、等周不等式、广义相对论和宇宙学中。尽管有大量关于密度流形的Ricci曲率的研究，但直到最近还没有相应的加权截面曲率理论；这个项目的另一个目标是进一步发展加权截面曲率界限理论，包括调查应用和与其他数学领域的联系。这位研究者最近的合作工作还引入了一种新的几何方法来研究具有密度的流形，该方法将某种扭转自由仿射联系作为基本的研究对象。这种方法不仅提供了对加权Ricci和截面曲率的新见解，而且还提出了新的自然结构，例如将被研究的加权完整群。

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.