CAREER: Gauge-theoretic Floer invariants, C* algebras, and applications of analysis to topology

该项目的主要研究目标是将来自物理学的分析工具，如规范理论和算子代数，应用于拓扑，这是对几何形状的研究。本研究分为两个主题：低维拓扑和算子k理论。在这两个领域中，上述分析工具都用于构建不变量来研究流形的几何结构，流形是基于欧几里德空间建模的空间，就像我们生活的三维空间一样。在低维拓扑和k算子理论中，PI将使用分析工具来研究这些空间的问题，比如它们是如何弯曲的，或者物体是如何嵌入其中的。这些问题在生物学和物理学中有着广泛的应用。该项目的教育和推广目标包括初中和高中阶段的数学和一般STEM丰富课程，重点是针对服务不足社区和代表性不足群体的学生的课程，以及高中、本科和研究生阶段的研究指导。在低维拓扑中，本项目着重于进一步理解瞬子和单极子Floer同调及其与Khovanov同调的关系，并以此研究流形上具有正标量曲率的度量族的存在性问题，以及结调和问题。另外，该项目还涉及计算研究结的一致性，既通过计算机搜索一致性，也通过计算研究某些局部等价和几乎局部等价群，这些群从结的一致性群中得到同态。在算子代数中，本项目主要研究算子代数的k理论及其在几何和拓扑不变量中的应用。算子代数的k理论群是椭圆算子的索引映射的目标，在流形的几何和拓扑中有着重要的应用。该项目涉及研究某些C*-代数的k理论，并利用它们来研究无限维空间；研究作用于这些无限维空间的群的非交换几何，特别是这些群的强Novikov猜想；研究高维膨胀机的粗糙Baum-Connes猜想。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The main research goal of this project is to apply analytic tools coming from physics, such as gauge theory and operator algebras, to topology, which is the study of geometric shapes. This research is divided into two themes: low dimensional topology and operator K-theory. In both fields, the aforementioned analytic tools are used to build invariants to study the geometric structure of manifolds, which are spaces modelled on Euclidean spaces, like the 3-dimensional space we live in. In both low dimensional topology and operator K-theory, the PI will use analytic tools to study questions about these spaces, such as how they are curved or how objects can be embedded inside them. These questions have a wide range of applications in biology and physics. The educational and outreach goals of this project involve math and general STEM enrichment programs at the middle and high school levels, with a focus on programs aimed at students from underserved communities and underrepresented groups, as well as mentorship in research at the high school, undergraduate and graduate levels.In low dimensional topology, this project focuses on furthering our understanding of instanton and monopole Floer homologies and their relation to Khovanov homology, and using this to study existence questions of families of metrics with positive scalar curvature on manifolds, as well as questions about knot concordance. Separately this project also involves computationally studying knot concordance, both by a computer search for concordances and by computationally studying certain local equivalence and almost local equivalence groups that receive homomorphisms from the knot concordance groups. In operator algebras, this project focuses on studying their K-theory and its applications to invariants in geometry and topology. The K-theory groups of operator algebras are the targets of index maps of elliptic operators and have important applications to the geometry and topology of manifolds. This project involves studying the K-theory of certain C*-algebras and using them to study infinite dimensional spaces; studying the noncommutative geometry of groups that act on these infinite dimensional spaces and, in particular, the strong Novikov conjecture for these groups; and studying the coarse Baum-Connes conjecture for high dimensional expanders.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.