Mathematical Sciences: K-Theory, C*-Algebras and Index Theory for Elliptic Operators

数学科学：椭圆算子的 K 理论、C* 代数和指数理论

• 批准号: 8803006
• 负责人: Jerome Kaminker
• 金额: $ 11.93万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8803006&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Theory; Algebras; Index

The central mathematical concept in this project is that of index. In its most basic manifestation, index has to do with the number of extra conditions that must be imposed on a solution to a differential equation in order to specify it unambiguously. A famous theorem of Atiyah and Singer asserts that this sort of index is given in the simplest case by a winding number (counting the number of times a path in the plane winds around the origin) and more generally by an analogous higher-dimensional topological invariant. The Atiyah-Singer Theorem and its numerous progeny all have the form: lefthand side (index of a differential operator) equals righthand side (something you can compute from the ingredients that go into the operator). Progress consists in achieving a more and more sophisticated understanding of what the righthand side really is. Nowadays, this involves geometry, topology, and functional analysis. The most advanced point of view, represented by Professors Kaminker and Ji among others, computes index in the K-theory of an appropriate algebra of operators. Classically, this would be commutative, an algebra of functions on the underlying manifold, but in the presence of additional structure a more complicated algebra is called for. The investigators will seek to understand the algebra and its K-theory better, and to extract new invariants for differential operators from elements of the K- groups.

这个项目的中心数学概念是 指数. 在其最基本的表现形式中，索引与 解决方案必须附加的一些额外条件， 一个微分方程，以便明确地说明它。 一 Atiyah和Singer的著名定理断言， 索引在最简单的情况下由卷绕数（计数）给出 平面中路径绕原点的次数） 更一般地，通过类似的高维拓扑 不变的 Atiyah-Singer定理及其众多的后代 都有这样的形式：左手边（微分的指数 运算符）等于右侧（您可以根据 进入操作者的成分）。 进展包括 对什么是越来越复杂的理解 右边是真的。 如今，这涉及到几何学， 拓扑学和功能分析。 以教授为代表的最先进的观点 Kaminker和Ji等人在K理论中计算指数， 一个合适的算子代数传统上，这将是 交换的，在基础流形上的函数的代数， 但是在存在附加结构的情况下， 需要代数。 调查人员将试图了解 代数及其K-理论更好，并提取新的 K-元素的微分算子的不变量 组