Topology, Noncommutative Geometry, and Mathematical Physics

这个研究项目结合了拓扑学、非交换几何和数学物理领域。本研究的重点之一是将拓扑和非交换几何应用于弦理论的对偶性，以了解不同时空背景下弦理论对偶性的拓扑约束，并了解对偶性导致非交换空间上的场论的情况。另一个主要的研究领域将是利用非交换几何和指数理论中的不变量来研究流形和类流形空间的几何和拓扑，例如具有奇点的流形。Rosenberg还将使用几何分析和微分拓扑的方法来研究流形上具有正标量曲率的黎曼度量空间的拓扑，包括低维和高维。最后，PI将继续准备课程材料，以便在二年级的多变量微积分和微分方程科学/工程课程中使用数学软件。数学（尤其是几何）和物理之间的相互作用对这两个学科来说都是一个长期而富有成果的学科。这个项目将推动这种相互作用，依靠物理学来提出有趣的几何研究课题，并利用我们对几何的理解来限制基本的物理理论。该项目的更广泛影响将包括(a)对研究生和本科生的指导，(b)为本科生、研究生和博士后的培训开发材料和课程，以及(c)向广大受众传播几何和数学物理的研究。

This research program combines the areas of topology, noncommutative geometry, and mathematical physics. One focus of this proposal will be on applications of topology and noncommutative geometry to duality in string theory, in order to understand topological constraints on dualities between string theories on different spacetime backgrounds, and also to understand situations where duality leads to a field theory on a noncommutative space. Another major area of investigation will be the use of invariants from noncommutative geometry and index theory for studying the geometry and topology of manifolds and of manifold-like spaces, such as manifolds with singularities. Rosenberg will also use methods of geometric analysis and differential topology to study the topology of the space of Riemannian metrics of positive scalar curvature on a manifold, in both low and high dimensions. Finally, the PI will continue preparation of course materials for the use of mathematical software in sophomore-level science/engineering courses in multivariable calculus and differential equations.The interaction between mathematics (especially geometry) and physics has been a long and fruitful one for both disciplines. This project will advance that interaction, by relying on physics to suggest interesting topics of study in geometry, and using our understanding of geometry to put restrictions on fundamental physical theories. Broader impacts of this project will include (a) mentoring of graduate and undergraduate students, (b) development of materials and courses for the training of undergraduates, graduate students and postdocs, and (c) dissemination of research in geometry and mathematical physics to a wide audience.