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

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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. ***