Mathematical Sciences: Algebraic Cycles and the Homotopy Theory of Groups

数学科学：代数圈和群的同伦论

• 批准号: 9400235
• 负责人: Eric Friedlander
• 金额: $ 31.3万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400235&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Algebraic; Cycles; Homotopy

9400235 Friedlander Friedlander plans to continue his investigation of algebraic cycles, using techniques from algebraic geometry and algebraic topology. The program of Friedlander and others holds promise in that it introduces a new perspective and imports techniques of algebraic topology. Several specific avenues of research appear ripe for further exploration: topological filtrations on algebraic cycles and homology, duality between Lawson homology and morphic cohomology, motivic complexes in the guise of algebraic cycle homology, and Chern classes in the context of algebraic cycle spaces. In addition to this study of algebraic cycles, Friedlander will continue his efforts to study the geometry implicit in the cohomology of infinitesimal algebraic groups. Priddy plans to continue his program to study the homotopy type of classifying spaces of groups and related constructions. Many of the most important questions in homotopy theory are related to classifying spaces of finite or compact Lie groups. Recently there have also developed interesting and powerful connections between topology and group theory, especially the cohomology and modular representation theory of finite groups. With the solution of the Segal and Sullivan Conjectures, this area has developed rapidly in recent years to the point where fundamental questions are being answered. Perhaps the most important of these is to determine the exact relationship between the stable and unstable homotopy types of a classifying space, completed at a prime p, and the p-local structure of its underlying group. Algebraic geometry is the study of solution sets of polynomial equations (i.e., algebraic varieties) using geometric techniques. Partial answers to questions in algebraic geometry have led to progress in fields ranging from complexity theory of computer science to geometric topology to number theory. Friedlander intends to study algebraic varieties, using methods borrowed from algebraic topology, as well as modern techniques of algebraic geometry. The use of topology involves the study of continuously varying families of structures, which have traditionally been considered by other means. The hope is that these new techniques will offer insight into deep and long-standing problems of algebraic geometry. Algebraic topology is the study of geometric objects by means of algebraic techniques. Exciting new developments have led to advances in group theory, using algebraic topology, thus reversing the direction of the usual flow of information. Groups are the fundamental symmetries occurring in all sciences, including areas involving codes, and structures in physics. Priddy hopes that this new approach will lead to a better understanding of the relationship between these fields. ***

9400235 Friedlander Friedlander计划使用代数几何和代数拓扑的技术继续对代数周期进行调查。 弗里德兰德（Friedlander）和其他人的计划有望，因为它引入了一种新的观点，并进口了代数拓扑的技术。 进一步探索的几个特定研究途径似乎已经成熟：关于代数周期和同源性的拓扑过滤，Lawson同源性和形态学的二元性，代数循环同源性幌子的动机复合物以及代数循环空间上下文中的Chern类。 除了对代数周期的研究外，弗里德兰德还将继续努力研究无限量代数群体的共同体中隐含的几何形状。 Priddy计划继续他的计划，研究组和相关结构的分类空间的同质类型。 同型理论中许多最重要的问题与有限或紧凑的谎言群体的空间进行分类有关。 最近，拓扑与群体理论之间也建立了有趣且有力的联系，尤其是有限群体的共同体和模块化代表理论。 通过解决Segal和Sullivan猜想的解决方案，近年来，该领域已经迅速发展，直到回答了基本问题。 也许其中最重要的是确定分类空间的稳定和不稳定的同型类型（在Prime P下完成）与其基础组的P-Local结构之间的确切关系。 代数几何形状是对使用几何技术的多项式方程（即代数品种）的溶液集的研究。 代数几何形状中问题的部分答案导致了从计算机科学的复杂性理论到几何拓扑结构到数字理论的领域进步。 弗里德兰德（Friedlander）打算使用从代数拓扑借用的方法以及代数几何技术研究代数品种。 拓扑的使用涉及研究连续变化的结构家族，这些结构传统上被其他方式考虑在内。 希望这些新技术能够深入了解代数几何的深层和长期存在的问题。 代数拓扑是通过代数技术对几何对象进行的研究。 令人兴奋的新事态发展导致了使用代数拓扑的群体理论的进步，从而扭转了通常的信息流的方向。 组是所有科学中发生的基本对称性，包括涉及代码的领域和物理结构。 Priddy希望这种新方法能够更好地了解这些领域之间的关系。 ***