K-theory of Operator Algebras and Invariants of Elliptic Operators

经典几何的一个基本思想是几何空间和坐标函数代数之间的对应关系。笛卡尔的非凡见解是，任何方程都会产生一个几何物体。这种见解得到了进一步的发展，首先是在代数几何中，后来发展到几何和拓扑学。对于经典几何空间，其坐标的乘法是可交换的，即乘法不依赖于运算的顺序。然而，在数学和物理学中，特别是量子力学中，有许多自然的“几何对象”，它们的坐标不再交换。非交换几何是数学的一个分支，专门用于处理这种非交换的“几何对象”。微分方程的不变量和更一般的算子代数的k理论是非交换几何的核心部分。主要研究一类微分方程的不变量，并将其应用于几何和拓扑问题的研究。主要研究者的长期研究目标是利用算子代数的k理论和更一般的非交换几何的方法来研究椭圆算子的高指标理论不变量及其在几何和拓扑中的应用。首席研究员能够使用这些方法获得各种几何应用，例如几何中的正标量曲率度量问题和拓扑中的流形刚性问题。在本项目中，主要研究者计划扩展这些方法以获得新的几何和拓扑应用，其中包括：(1)给定自旋流形上正标量曲率度量空间的高同伦群；(2)奇异空间的外科理论；(3)奇异变分的Grothendieck-Riemann-Roch定理；(4)定量代数k理论及其应用。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。

A fundamental idea in classical geometry is the correspondence between geometric spaces and algebras of coordinate functions. Descartes' remarkable insight was that any equation gives rise to a geometric object. This insight was further developed, first notably in algebraic geometry, later progressing to geometry and topology. For a classical geometric space, the multiplication of its coordinates is commutative, namely, the multiplication does not depend on the order of operation. However, there are many natural "geometric objects" in both mathematics and physics, especially quantum mechanics, where the coordinates no longer commute. Noncommutative geometry is a branch of mathematics that is designed to handle just such noncommutative "geometric objects". Invariants of differential equations and more generally K-theory of operator algebras are a central part of noncommutative geometry. The principal investigator intends to study a certain class of invariants of differential equations and apply them to study problems in geometry and topology.A long term research goal of the principal investigator is to use methods from K-theory of operator algebras and more generally noncommutative geometry to study higher index theoretic invariants of elliptic operators and their applications to geometry and topology. The principal investigator was able to use these methods to obtain various geometric applications such as in the case of positive scalar curvature metrics problems in geometry and in the case of the manifold rigidity problem in topology. In this project, the principal investigator plans to extend these methods to obtain new geometric and topological applications, which include (1) the higher homotopy groups of the space of positive scalar curvature metrics on a given spin manifold; (2) surgery theory of singular spaces; (3) the Grothendieck-Riemann-Roch theorem for singular varieties; (4) quantitative algebraic K-theory and its applications.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.