K-theories, Cycle Theories, and Cohomology Calculations

Northwestern University, Evanston IL

Investigators

Abstract

Friedlander-Abstract The relationship between algebraic K-theory and algebraic cycles is of fundamental importance. Friedlander proposes to consider his investigations of this relationship by continuing to contemplate the spectral sequence developed in collaboration with Suslin and by studying Chern classes to various cohomology theories. Much of Friedlander's effort will be directed to semi-topological K-theory (being developed in collaboration with Mark Walker) and its relationship with morphic cohomology (developed in collaboration with Blaine Lawson). Friedlander also proposes to further investigate the representation theory of algebras related to algebraic groups, in part through the thesis work of several Ph.D. students. Throughout the twentieth century, algebraic topology (mathematics of "bending and twisting") and algebraic geometry (mathematics of "surfaces" coming from graphing polynomial equations) enjoyed a very positive interactive relationship. The types of geometric objects ("surfaces") studied by topologists can be more general than geometric "surfaces" called algebraic varieties, but many of the most basic and interesting topological "surfaces" are algebraic varieties. Constructions and techniques arising naturally in either geometry or topology continue to be successfully imported to the other area, often with surprising computational or conceptual results. Much of the proposed work is to use techniques and to pose questions that are topological in character to gain a better understanding of some central aspects of algebraic geometry.