Mathematical Sciences: Torsion Invariants for Commuting Systems of Operators

数学科学：算子通勤系统的扭转不变量

• 批准号: 9502154
• 负责人: Richard Carey
• 金额: $ 7.15万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9502154&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Torsion; Invariants; Commuting

9502154 Carey The investigator is studying operator algebras. More specifically, he is studying what are known as Steinberg symbols, a generalization of multiplicative commutators in such algebras. Steinberg symbols are factored by means of a Koszul complex construction, and this gives rise to a new invariant, the joint torsion. This invariant is related to a local or maximal ideal index in the setting of commutative operator algebras, and it provides new understanding of the principal function in the non- commutative case. Similar constructions involving higher algebraic K-groups will also be investigated. The study of Wiener-Hopf integral equations began more than sixty years ago in connection with physical problems such as the diffraction of electromagnetic or sound waves. A natural operator W arises. The investigator's contributions began with the introduction of a naturally paired operator U for which a combination of the two operators known as the determinant took on a particularly simple and useful integral form. Using this determinant, he found that the study of the Wiener-Hopf operator W could be deepened, that new solutions could be found to the original integral equations, and that the study of functions h(W,U) of the pair of operators could be put into a new geometric context. This was an early step in development of the hugely successful new area of modern mathematics sometimes known as "non-commutative geometry." Now non-commutative geometry has drawn together several diverse areas of mathematics, and "algebraic K-theory" figures prominently among them. In the context of algebraic K-theory, the further analysis of the structure of the determinants mentioned above has led to a new object called the joint torsion of the pair {W,U}. This object in turn is linked to geometry in still another way and occurs in the study of differential equations and in the study of dynamical systems. ***

小行星9502154 研究者正在研究算子代数。 更具体地说，他正在研究所谓的斯坦伯格符号，在这样的代数乘法算子的推广。 斯坦伯格符号的分解通过一个Koszul复杂的建设，这引起了一个新的不变量，联合扭转。 这个不变量与交换算子代数中的一个局部或极大理想指标有关，它提供了对非交换情形下主函数的新理解。 类似的建设涉及更高的代数K-群也将进行调查。 维纳-霍普夫积分方程的研究始于六十多年前，与电磁波或声波衍射等物理问题有关。 自然算子W产生。 调查员的贡献开始介绍了一个自然配对运营商U的组合的两个运营商被称为行列式采取了特别简单和有用的积分形式。 使用这个行列式，他发现，研究维纳-霍普夫算子W可以深化，新的解决方案可以找到原来的积分方程，并研究功能h（W，U）的对运营商可以投入到一个新的几何背景。 这是现代数学中取得巨大成功的新领域--有时被称为“非对易几何”--发展的早期步骤。现在，非对易几何已经把几个不同的数学领域联系在一起，而“代数K理论”在其中占有突出地位。 在代数K-理论的背景下，对上述行列式结构的进一步分析导致了一个新的对象，称为对{W，U}的联合扭转。 这个对象反过来又与几何学以另一种方式，并发生在研究微分方程和研究动力系统。 ***