Algebraic Cycles, K-Theory, and Representation Theory

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DMS-0300525Eric M. FriedlanderFriedlander proposes to investigate topics in algebra, geometry,and topology. Each of these topics entail a synthesis of techniquesand results from various mathematical fields with the goal of progresstoward solutions of fundamental problems, and each has seen progressachieved by Friedlander and his collaborators. Firstly, Friedlanderproposes to investigate algebraic K-theory and algebraic cycles onalgebraic varieties, with the expectation that his investigation willcontribute both specific computations and general properties of thesefundamental invariants. Friedlander will seek to produce topologicalconstructions associated to objects arising in algebraic geometrywhich closely reflect subtle aspects of algebraic cycles and algebraicK-theory. These constructions, many planned in conjunction with MarkWalker, are envisioned to involve a blend of techniques from stablehomotopy theory and recent techniques developed by Voevodsky for motiviccohomology. In particular, Friedlander plans to investigate further thesemi-topological K-theory of varieties and its connections with algebraicand topological K-theory. The second topic involves the introductionof new spaces determined by the representation theory of a finite groupscheme which provide a new perspective on cohomological support varieties.The goal of this research, in part to be achieved in collaboration withJulia Pevtsova, is to produce finer invariants in the general context offinite group schemes which are accessible to computations and which extendour understanding of (modular) representations. Finally, in joint workwith Vincent Franjou, Friedlander proposes to study the cohomology ofpolynomial bifunctors with the aim of improving earlier computations byhimself and others to cases more closely related to questions in K-theory.Mathematics continues to reveal beautiful relationships which are bothuseful 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 overthe centuries. Some of the questions considered still seem dauntinglydifficult, but partial progress towards their solutions will lead toadvances in different branches of mathematics and mathematical physics.A second aspect of this project is the study of formal algebraic objectswhich arise as symmetries of familiar structures. Once again, geometryis blended with algebra to provide motivation for questions to be askedas well as to suggest methods of solution. A third aspect consists ofefforts to maintain the strength of the national effort in mathematics bymentoring graduate students and junior faculty, by organizing mathematicalmeetings, by editorial efforts for journals and special volumes, andby participation in the on-going discussion of policy issues for theAmerican Mathematical Society.