Topology, Noncommutative Geometry, and Mathematical Physics

这项研究项目探索了数学中的几何和物理中的弦理论之间的接口问题。该项目旨在通过使用非对易几何和K-理论等数学领域的方法来研究物理理论的结构，并反过来利用物理的思想来研究几何和拓扑中的新现象。这一结果有望推进数学和物理领域的几个研究领域。研究生和本科生将通过参与跨学科项目接受研究培训，研究人员将继续积极领导K-理论社区。该项目的三个主要重点领域是拓扑K-理论与弦理论的关系，非对易流形的几何，以及正标量曲率的度量和Yamabe不变量的研究。在第一个领域，该项目将研究K-理论如何约束和建议弦理论、F-理论和M-理论中的对偶，以及为什么与T-对偶相关的K-理论同构似乎与Baum-Connes猜想有关。在第二个领域，该项目旨在了解非交换2-环面上非共形平坦的度量的几何，并证明关于此类度量的一个非交换Gauss-Bonnet定理。在最后一个领域，该项目将尝试建立什么曲率正性条件保证Witten亏格中的项的消失，并更好地理解流形上正的标量曲率度量空间的拓扑。

This research project explores questions at the interface of geometry in mathematics and string theory in physics. The project aims to investigate the structure of physical theories by using methods from the mathematical fields of non-commutative geometry and K-theory, and in turn, to utilize ideas from physics to study new phenomena in geometry and topology. The results are expected to advance several research areas in mathematics and physics. Graduate and undergraduate students will receive research training through involvement in the interdisciplinary project, and the investigator will continue active leadership in the K-theory community.The three primary areas of focus for the project are the relationship between topological K-theory and string theory, the geometry of noncommutative manifolds, and the study of metrics of positive scalar curvature and the Yamabe invariant. In the first area, the project will study how K-theory constrains and suggests dualities in string theory, F-theory, and M-theory, and why K-theory isomorphisms associated with T-dualities seem to be related to the Baum-Connes conjecture. In the second area, the project aims to understand the geometry of metrics on the noncommutative 2-torus that are not conformally flat, and to prove a noncommutative Gauss-Bonnet theorem for such metrics. In the last area, the project will attempt to establish what curvature positivity conditions guarantee vanishing of terms in the Witten genus, and to better understand the topology of the space of positive scalar curvature metrics on a manifold.