Mathematical Sciences: Algebraic Cycles, Group Schemes, K-Theory and Connections between Stable Homotopy and Group Cohomology

数学科学：代数环、群方案、K 理论以及稳定同伦与群上同调之间的联系

• 批准号: 9704794
• 负责人: Eric Friedlander
• 金额: $ 18.54万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704794&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Algebraic; Cycles; Group

9704794 Friedlander Eric M. Friedlander will study algebraic cycles from two points of view: the first is more topological and differential-geometric in nature, whereas the second is more algebraic and is allied with the study of motivic cohomology. Particular attention will be given to relationships with algebraic K-theory, long-standing problems in algebraic geometry, and the challenge of producing concrete examples. Moreover, the study of representations of various Hopf algebras associated to algebraic groups will continue. Of special interest are the representations of infinitesimal group schemes. Stewart B. Priddy will study connections between stable homotopy theory and the cohomology and representation theory of groups. The goal is to develop a global point of view by working at the interface of these fields, using insights from the stable homotopy theory of classifying spaces. One avenue of research is the study of the stable type of classifying spaces of products of groups. Another is the study of the representation theory conjectures of Alperin and Broue', which may be amenable to equivariant topological methods. One of the principal techniques is the use of stable decompositions of classifying spaces to suggest concepts in cohomology of groups, as was the case with Swan groups. Aspects of Friedlander's research relate to fundamental problems in mathematics whose history can be traced back to the previous century. Many central problems of algebraic geometry, especially those involving algebraic cycles, have challenged generations of mathematicians. New techniques from topology and formal algebraic geometry hold promise of at least partial solutions. In a different mathematical arena, the classical representation theory of algebraic groups is reflected by the representations of related algebraic structures. The interplay between the classical theory and the latest techniques involving categorical constructions presents opportunities for i mportant developments. In contrast with Priddy's emphasis, the field of algebraic topology has long sought, with much success, to apply algebraic methods to the solution of geometric or topological problems. By reformulating topological problems in algebraic terms, one is often able to answer a more tractable question. In general, however, there are many connections between these areas, which enables one to reverse this procedure and bring topological methods to bear in algebra. Priddy's project studies the interaction between topology and group theory, with the aim of uncovering fundamental algebraic structures. Powerful symbolic manipulation techniques using high speed computers comprise part of the project. ***

9704794 Friedlander Eric M. Friedlander将从两个角度研究代数循环：第一个本质上是拓扑和差异几何，而第二个则更为代数，而第二个代数则与动机同学的研究相关。 特别关注与代数K理论的关系，代数几何学的长期问题以及产生具体示例的挑战。 此外，将继续对与代数群体相关的各种HOPF代数的表示。 特别感兴趣的是无穷小组计划的表示。 Stewart B. Priddy将研究稳定的同型理论与群体的同步理论之间的联系。 目的是利用稳定的分类空间同义理论的见解来通过在这些字段的界面上工作来发展全球观点。 研究的一种途径是研究稳定类型的群体产品分类空间。 另一个是对Alperin和Broue'的表示理论猜想的研究，这可能适合于等效性拓扑方法。 主要技术之一是使用稳定的分解空间的分类来提出群体共同体中的概念，就像天鹅群体一样。 弗里德兰德（Friedlander）研究的各个方面涉及数学中的基本问题，其历史可以追溯到上个世纪。 代数几何形状，尤其是涉及代数周期的几何形状的许多核心问题，都挑战了几代数学家。 拓扑和正式代数几何形状的新技术至少有希望的部分解决方案。 在不同的数学领域中，代数群的经典表示理论反映了相关代数结构的表示。 经典理论与涉及分类结构的最新技术之间的相互作用为我的发展提供了机会。 与Priddy的重点相反，代数拓扑领域长期以来一直在寻求将代数方法应用于几何或拓扑问题的解决方案。 通过用代数术语重新制定拓扑问题，人们通常能够回答一个更容易解决的问题。 但是，通常，这些区域之间存在许多联系，这使人们可以扭转此过程并将拓扑方法带入代数。 Priddy的项目研究拓扑与群体理论之间的相互作用，目的是揭示基本代数结构。 使用高速计算机的强大符号操作技术包括项目的一部分。 ***