Topology, Noncommutative Geometry, and Mathematical Physics

这个研究计划结合了拓扑学，非交换几何和数学物理领域。这个计划的一个重点是将拓扑学和非对易几何应用于弦论中的对偶性，以理解不同时空背景下的弦论之间对偶性的拓扑约束，并理解对偶性导致非对易空间上的场论的情况。另一个主要的研究领域将是使用不变量从非交换几何和指数理论研究几何和拓扑的流形和流形样空间，如流形与奇点。罗森博格还将使用几何分析和微分拓扑的方法来研究拓扑空间的黎曼度量的正标量曲率的流形上，在低和高的层面。最后，PI将继续编写课程材料，以便在多变量微积分和微分方程的大学级科学/工程课程中使用数学软件。数学（特别是几何）和物理之间的相互作用对这两个学科来说都是长期而富有成果的。 这个项目将通过依靠物理学来提出有趣的几何研究主题，并利用我们对几何的理解来限制基本物理理论，从而促进这种相互作用。该项目的更广泛影响将包括：（a）指导研究生和本科生;（B）编制培训本科生、研究生和博士后的材料和课程;（c）向广大受众传播几何和数学物理研究成果。

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.