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

小行星9704794 Eric M.弗里德兰德将研究代数圈从两个角度来看：第一个是更多的拓扑和微分几何的性质，而第二个是更多的代数和结盟的研究motivic上同调。 将特别注意与代数K-理论，在代数几何长期存在的问题，并产生具体的例子的挑战的关系。 此外，与代数群相关的各种Hopf代数的表示的研究将继续下去。 特别感兴趣的是无穷小群方案的表示。 斯图尔特B。Priddy将研究稳定同伦理论和上同调和群的表示理论之间的联系。 我们的目标是通过在这些领域的接口工作，使用来自分类空间的稳定同伦理论的见解，发展一个全球性的观点。 研究的一个途径是研究群的乘积的分类空间的稳定型。 另一个是研究Alperin和Broue的表示论结构，这可能适用于等变拓扑方法。 其中一个主要的技术是使用稳定分解的分类空间，以建议概念的上同调的群体，是与天鹅组的情况。 方面弗里德兰德的研究涉及到基本问题的数学，其历史可以追溯到上世纪。 代数几何的许多中心问题，特别是那些涉及代数圈的问题，已经挑战了几代数学家。 拓扑学和形式代数几何学的新技术至少可以部分解决这个问题。 在另一个不同的数学竞技场中，代数群的经典表示理论被相关代数结构的表示所反映。 经典理论与范畴建构的最新技术之间的相互作用为重要的发展提供了机会。 在对比普里迪的重点，该领域的代数拓扑长期以来一直寻求，并取得了很大的成功，应用代数方法的解决几何或拓扑问题。 通过用代数术语重新表述拓扑问题，人们往往能够回答一个更易处理的问题。 然而，一般来说，这些领域之间有许多联系，这使人们能够扭转这一过程，并将拓扑方法应用于代数。 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操作技术包括该项目的一部分。 ***

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. ***