Topology, Noncommutative Geometry, and Mathematical Physics

这个研究项目探讨了数学中的几何和物理中的弦理论的接口问题。该项目旨在利用非交换几何和k理论等数学领域的方法来研究物理理论的结构，进而利用物理学的思想来研究几何和拓扑中的新现象。这一结果有望推动数学和物理学的几个研究领域。研究生和本科生将通过参与跨学科项目接受研究培训，研究者将继续在k理论社区中发挥积极的领导作用。该项目的三个主要关注领域是拓扑k理论和弦理论之间的关系，非交换流形的几何，以及正标量曲率和Yamabe不变量的度量研究。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理论中的对偶性，以及为什么与t对偶性相关的k理论同构似乎与Baum-Connes猜想有关。在第二个领域，该项目旨在了解非共形平坦的非交换2环面上的度量几何，并证明此类度量的非交换高斯-邦尼特定理。在最后一个领域，该项目将尝试建立什么样的曲率正性条件保证Witten格中的项消失，并更好地理解流形上正标量曲率度量空间的拓扑结构。

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.