K-theories, Cycle Theories, and Cohomology Calculations

代数K-理论和代数圈之间的关系是非常重要的。 弗里德兰德建议考虑他的调查，这种关系，继续考虑频谱序列开发合作与苏斯林和研究陈类各种上同调理论。 弗里德兰德的大部分工作将致力于半拓扑K理论（与马克步行者合作开发）及其与形态上同调的关系（与布莱恩·劳森合作开发）。 Friedlander还建议进一步研究与代数群有关的代数的表示理论，部分是通过几位博士的论文工作。在整个二十世纪，代数拓扑学（数学的“弯曲和扭曲”）和代数几何学（数学的“表面”来自图形多项式方程）享有一个非常积极的互动关系。 拓扑学家所研究的几何对象（“曲面”）的类型可以比称为代数簇的几何“曲面”更一般，但许多最基本和最有趣的拓扑“曲面”都是代数簇。 在几何学或拓扑学中自然产生的构造和技术继续成功地导入到其他领域，通常具有令人惊讶的计算或概念结果。 大部分拟议的工作是使用技术和提出的问题是拓扑性质，以获得更好地了解一些核心方面的代数几何。

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.