Mathematical Sciences: Algebraic Cycles and the Homotopy Theory of Groups

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9400235 Friedlander Friedlander plans to continue his investigation of algebraic cycles, using techniques from algebraic geometry and algebraic topology. The program of Friedlander and others holds promise in that it introduces a new perspective and imports techniques of algebraic topology. Several specific avenues of research appear ripe for further exploration: topological filtrations on algebraic cycles and homology, duality between Lawson homology and morphic cohomology, motivic complexes in the guise of algebraic cycle homology, and Chern classes in the context of algebraic cycle spaces. In addition to this study of algebraic cycles, Friedlander will continue his efforts to study the geometry implicit in the cohomology of infinitesimal algebraic groups. Priddy plans to continue his program to study the homotopy type of classifying spaces of groups and related constructions. Many of the most important questions in homotopy theory are related to classifying spaces of finite or compact Lie groups. Recently there have also developed interesting and powerful connections between topology and group theory, especially the cohomology and modular representation theory of finite groups. With the solution of the Segal and Sullivan Conjectures, this area has developed rapidly in recent years to the point where fundamental questions are being answered. Perhaps the most important of these is to determine the exact relationship between the stable and unstable homotopy types of a classifying space, completed at a prime p, and the p-local structure of its underlying group. Algebraic geometry is the study of solution sets of polynomial equations (i.e., algebraic varieties) using geometric techniques. Partial answers to questions in algebraic geometry have led to progress in fields ranging from complexity theory of computer science to geometric topology to number theory. Friedlander intends to study algebraic varieties, using methods borrowed from algebraic topology, as well as modern techniques of algebraic geometry. The use of topology involves the study of continuously varying families of structures, which have traditionally been considered by other means. The hope is that these new techniques will offer insight into deep and long-standing problems of algebraic geometry. Algebraic topology is the study of geometric objects by means of algebraic techniques. Exciting new developments have led to advances in group theory, using algebraic topology, thus reversing the direction of the usual flow of information. Groups are the fundamental symmetries occurring in all sciences, including areas involving codes, and structures in physics. Priddy hopes that this new approach will lead to a better understanding of the relationship between these fields. ***