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

9704794 Friedlander Eric M.Friedlander将从两个角度研究代数圈：第一个更具拓扑性和微分几何性质，而第二个更具代数性并与基元上同调的研究相结合。将特别关注与代数K理论的关系、代数几何中的长期存在的问题，以及产生具体例子的挑战。此外，还将继续研究与代数群相关的各种Hopf代数的表示。特别感兴趣的是无穷小群方案的表示。斯图尔特·B·普里迪将研究稳定同伦理论与群的上同调和表示理论之间的联系。我们的目标是通过在这些领域的界面上工作，利用来自分类空间的稳定同伦理论的见解，发展一种全球观点。研究的一个途径是研究群乘积的分类空间的稳定型。另一类是研究Alperin和Broue的表象理论猜想，这些猜想可能适用于等变拓扑学方法。主要技术之一是使用分类空间的稳定分解来提出群的上同调的概念，就像Swan群的情况一样。弗里德兰德的研究涉及数学中的基本问题，其历史可以追溯到上个世纪。代数几何的许多中心问题，特别是涉及代数圈的问题，挑战了一代又一代的数学家。拓扑学和形式代数几何的新技术保证了至少部分解的存在。在一个不同的数学领域，代数群的经典表示理论被相关的代数结构的表示所反映。经典理论和涉及范畴结构的最新技术之间的相互作用为重要的发展提供了机会。与Priddy强调的相反，代数拓扑学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，并取得了很大成功。通过用代数重新表述拓扑学问题，人们通常能够回答一个更容易处理的问题。然而，总的来说，这些领域之间有许多联系，这使得人们能够逆转这一过程，并将拓扑学方法应用于代数。Priddy的项目研究拓扑学和群论之间的相互作用，目的是揭示基本的代数结构。使用高速计算机的强大的符号处理技术构成了该项目的一部分。***

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. ***