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

职业：规范理论 Floer 不变量、C* 代数以及拓扑分析应用

• 批准号: 2340465
• 负责人: Sherry Gong
• 金额: $ 54.93万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-09-01 至 2029-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2340465&HistoricalAwards=false
• 关键词: CAREER; Gauge; theoretic; Floer; invariants

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.

该项目的主要研究目标是将来自物理学的分析工具（例如仪表理论和操作员代数）应用于拓扑，这是对几何形状的研究。这项研究分为两个主题：低维拓扑和操作员K理论。在这两个领域中，上述分析工具都用于构建不变性来研究歧管的几何结构，这些空间是在欧几里得空间上建模的空间，例如我们居住的三维空间。在低维拓扑和操作员K-theor中，PI都将使用这些空间来研究这些空间的问题，例如它们在这些空间中的质疑，例如它们的范围或对象又可以对象进行对象。这些问题在生物学和物理学中具有广泛的应用。 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,并利用这一点来研究具有积极标态曲率的指标家庭的存在问题，以及有关结一致性的问题。该项目分别涉及计算研究结的一致性，无论是通过计算机搜索一致性，以及通过计算研究某些局部等效性和几乎局部等效组，这些群体从结的一致性组中接受同构。在操作员代数中，该项目致力于研究其K理论及其在几何和拓扑中不变的应用。操作员代数的K理论组是椭圆运算符的索引图的目标，并且在歧管的几何形状和拓扑中具有重要的应用。该项目涉及研究某些C* - 代数的K理论，并使用它们来研究无限的维空间。研究对这些无限的维空间，尤其是这些群体强烈的诺维科夫猜想的群体的非交通性几何形状；并研究了对高维扩展器的粗糙鲍姆 - 康涅狄格州的猜想。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响标准，被认为值得通过评估来获得支持。