Algebraic Cycles, K-Theory, and Representation Theory

DMS-0300525 Eric M. FriedlanderFriedlander建议调查代数，几何和拓扑学的主题。 每一个主题都需要综合各种数学领域的技术和结果，目标是解决基本问题，每个主题都看到了Friedlander和他的合作者取得的进展。 首先，Friedlander提出研究代数簇上的代数K-理论和代数圈，期望他的研究将有助于这些基本不变量的具体计算和一般性质。 弗里德兰德将寻求产生topologicalconstructions相关的对象所产生的代数geometrywhich密切反映微妙方面的代数周期和algebraicK-理论。 这些建设，许多计划与马克沃克，设想涉及混合技术从stablehomotopy理论和最近的技术开发的Voevodsky motiviccohomology。 特别是，弗里德兰德计划进一步调查thesemi-topological K-理论的品种和它的连接与algebraicand拓扑K-理论。 第二个主题涉及到introductionof新的空间所确定的表示理论的有限群scheme提供了一个新的视角上同调支持variety.The本研究的目标，部分将实现与朱莉娅Pevtsova合作，是产生更精细的不变量的一般上下文中的有限群计划，可访问的计算和扩展我们的理解（模块化）表示。 最后，在与文森特Franjou联合工作，Friedlander提出研究多项式双函子的上同调，目的是改进自己和其他人的早期计算更密切相关的问题，在K理论的情况下，数学继续揭示美丽的关系，这是既有用又令人惊讶。 这个项目涉及的形状（拓扑学）的研究出现的多项式方程的解决方案。这样的研究使用几何的见解和代数操作，由许多数学家在过去几个世纪的建设和计算增强。 有些问题似乎仍然令人生畏的困难，但部分进展，对他们的解决方案将导致进步的不同分支的数学和数学物理。第二个方面，这个项目是研究正式的代数对象出现的对称性熟悉的结构。 再一次，几何与代数相结合，为提出问题提供了动力，并提出了解决问题的方法。 第三个方面包括ofecourse保持实力的国家努力在数学辅导研究生和初级教师，通过组织学术会议，编辑工作的期刊和特别卷，并通过参与正在进行的讨论的政策问题为theAmerican Mathematical Society。

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.