Quantitative Operator K-theory and Applications

在经典几何中，人们研究坐标互换的几何对象。非对易几何是一门专门为研究“几何对象”而设计的数学理论，这些“几何对象”的坐标不交换，但在数学和物理中是自然出现的。在过去的十年左右，在非对易几何的新思想的帮助下，在解决经典几何、拓扑学和数学物理中长期存在的问题方面取得了很大的进展。K-理论是非对易几何、经典几何、拓扑学和数学物理之间的桥梁。首席研究员和他的学生计划开发一种量化算子K-理论来研究这些数学领域中出现的问题。这个项目将通过指导研究生和博士后学生来促进美国劳动力的发展。主要研究人员和他的学生将专注于定量算子K-理论在几何、拓扑、离散群的几乎表示理论和算子代数的结构理论中的应用。特别是，研究人员打算应用量化算子K-理论技术来解决微分算子的高指数理论、标量曲率的量化行为、几乎表示的范数稳定性以及C*-代数结构理论中的泛系数问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

In classical geometry, one studies geometric objects whose coordinates commute. Noncommutative geometry is a mathematical theory specifically designed to study "geometric objects" whose coordinates do not commute but which do occur naturally in mathematics and physics. In the last decade or so, with the help of new ideas from noncommutative geometry, great advances have been made toward the solutions of long-standing problems in classical geometry, topology, and mathematical physics. K-theory serves as a bridge between noncommutative geometry, classical geometry, topology, and mathematical physics. The principal investigator and his students plan to develop a quantitative operator K-theory to investigate problems arising from these areas of mathematics. This project will contribute to US workforce development through mentoring of graduate and post-doctoral students.The principal investigator and his students will focus on the applications of quantitative operator K-theory to problems in geometry, topology, almost representation theory of discrete groups, and structure theory of operator algebras. In particular, the investigator intends to apply quantitative operator K-theory techniques to solve problems in higher index theory of differential operators, quantitative behavior of scalar curvature, norm stability of almost representations, and the universal coefficient problem in structure theory of C*-algebras.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.