Algebraic Cycles, K-Theory, and Representation Theory

Northwestern University, Evanston IL

Investigators

Abstract

DMS-0300525 Eric M. Friedlander Friedlander proposes to investigate topics in algebra, geometry, and topology. Each of these topics entail a synthesis of techniques and results from various mathematical fields with the goal of progress toward solutions of fundamental problems, and each has seen progress achieved by Friedlander and his collaborators. Firstly, Friedlander proposes to investigate algebraic K-theory and algebraic cycles on algebraic varieties, with the expectation that his investigation will contribute both specific computations and general properties of these fundamental invariants. Friedlander will seek to produce topological constructions associated to objects arising in algebraic geometry which closely reflect subtle aspects of algebraic cycles and algebraic K-theory. These constructions, many planned in conjunction with Mark Walker, are envisioned to involve a blend of techniques from stable homotopy theory and recent techniques developed by Voevodsky for motivic cohomology. In particular, Friedlander plans to investigate further the semi-topological K-theory of varieties and its connections with algebraic and topological K-theory. The second topic involves the introduction of new spaces determined by the representation theory of a finite group scheme which provide a new perspective on cohomological support varieties. The goal of this research, in part to be achieved in collaboration with Julia Pevtsova, is to produce finer invariants in the general context of finite group schemes which are accessible to computations and which extend our understanding of (modular) representations. Finally, in joint work with Vincent Franjou, Friedlander proposes to study the cohomology of polynomial bifunctors with the aim of improving earlier computations by himself and others to cases more closely related to questions in K-theory. Mathematics continues to reveal beautiful relationships which are both useful and surprising. This project involves the study of shapes (topology) which arise as the solutions of polynomial equations. Such a study uses geometric insights and algebraic manipulations, augmented by constructions and computations of many mathematicians over the centuries. Some of the questions considered still seem dauntingly difficult, but partial progress towards their solutions will lead to advances in different branches of mathematics and mathematical physics. A second aspect of this project is the study of formal algebraic objects which arise as symmetries of familiar structures. Once again, geometry is blended with algebra to provide motivation for questions to be asked as well as to suggest methods of solution. A third aspect consists of efforts to maintain the strength of the national effort in mathematics by mentoring graduate students and junior faculty, by organizing mathematical meetings, by editorial efforts for journals and special volumes, and by participation in the on-going discussion of policy issues for the American Mathematical Society.