K-theories, Cycle Theories, and Cohomology Calculations

摘要代数K-理论与代数圈之间的关系是非常重要的。弗里德兰德建议，通过继续思考与苏斯林合作开发的谱序列，并通过研究各种上同调理论的陈类，来考虑他对这种关系的研究。Friedlander的大部分工作将指向半拓扑K-理论(与Mark Walker合作开发)及其与态上同调(与Blaine Lawson合作开发)的关系。弗里德兰德还建议进一步研究与代数群有关的代数的表示理论，部分是通过几名博士生的论文工作。在整个20世纪，代数拓扑学(弯曲和扭转的数学)和代数几何(通过绘制多项式方程来表示曲面的数学)享有非常积极的互动关系。拓扑学家研究的几何对象(曲面)的类型可以比称为代数族的几何“曲面”更一般，但许多最基本和有趣的拓扑“曲面”是代数族。几何学或拓扑学中自然产生的构造和技术继续被成功地引入到另一个领域，通常会产生令人惊讶的计算或概念结果。许多建议的工作是使用技术和提出具有拓扑性的问题，以更好地理解代数几何的一些核心方面。

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.