Mathematical Sciences: Cyclic Homology, Algebra, Topology and Geometry

数学科学：循环同调、代数、拓扑和几何

• 批准号: 8701125
• 负责人: Dan Burghelea
• 金额: --
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-15 至 1989-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701125&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Cyclic; Homology; Algebra

Burghelea will study some aspects and implications of cyclic homology in algebra, topology and geometry. In algebra, he plans to study the cyclic homology of algebraic varieties and crossed product algebras associated to actions of groups on algebraic varieties; he will also study the Chern character for group rings. In topology he will study the homology type of B Diff(M) in the stability range, and at the prime 2. In geometry he will study the relationship between the Selberg trace formula, cyclic homology and closed geodesics. Exploring the uses of cyclic homology will render a powerful technique more versatile. Ultimately applications to theoretical physics and other consumers of higher mathematics should profit. **8 8701643 Bergman Three important properties of universal algebras will be studied: the amalgamation property, deductiveness, and congruence modular varieties. The research on the amalgamation property centers on two problems. 1. Determination if the amalgamation property plus residual smallness implies the congruence extension property. 2. Characterization of the amalgamation bases of the variety. The plans for deductiveness are less specific. The general issue is to attempt to classify those varieties that are deductive. A third avenue to be pursued is the representation problem for the commutator in congruence modular varieties. All of these topics concern the foundations of algebra and are likely to be applied outside of algebra only to the extent that they condition a mode of thought about algebraic questions.

Burghelea将研究循环同调在代数、拓扑和几何中的一些方面和含义。在代数，他计划研究循环同源的代数品种和交叉产品代数相关的行动组代数品种;他还将研究陈德铭字符组环。在拓扑学，他将研究的同源类型的B Diff（M）的稳定范围内，并在总理2。在几何，他将研究之间的关系塞尔伯格迹公式，循环同源和封闭测地线。探索循环同源性的用途将使一种强大的技术更加通用。最终，理论物理和其他高等数学消费者的应用应该会受益。**8 8701643伯格曼将研究泛代数的三个重要性质：合并性质，演绎性和同余模簇。对合并财产的研究主要集中在两个方面。1.合并性质加剩余小性是否蕴涵同余扩张性质的判定。2.品种融合基础的表征。演绎法的计划没有那么具体。一般的问题是试图对那些演绎的变体进行分类。第三个途径是追求的代表性问题的交换子在同余模品种。所有这些主题都涉及代数的基础，并且很可能在代数之外应用，只有在它们制约关于代数问题的思维模式的程度上。