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. ***

小行星9400235 弗里德兰德计划继续他的调查代数周期，使用技术从代数几何和代数拓扑。 该计划的弗里德兰德和其他人持有的承诺，因为它引入了一个新的角度和进口技术的代数拓扑。 几个具体途径的研究出现成熟的进一步探索：拓扑filtrations代数循环和同源性，劳森同调和形态上同调之间的对偶性，动机复合物的幌子代数循环同源性，陈类的背景下，代数循环空间。 除了这项研究的代数周期，弗里德兰德将继续努力研究几何隐含在上同调的无穷小代数群。 Priddy计划继续他的计划，研究同伦类型的分类空间的群体和相关的建设。 同伦理论中许多最重要的问题都与有限或紧李群空间的分类有关。 最近也有发展有趣和强大的连接拓扑和群论，特别是上同调和有限群的模表示理论。 随着Segal猜想和Sullivan猜想的提出，这一领域近年来发展迅速，基本问题得到了解答。 也许其中最重要的是确定分类空间的稳定和不稳定同伦类型之间的确切关系，在素数p处完成，以及其基础群的p-局部结构。 代数几何是研究多项式方程的解集（即，代数簇）使用几何技术。 对代数几何问题的部分解答导致了从计算机科学的复杂性理论到几何拓扑学再到数论等领域的进步。 弗里德兰德打算研究代数品种，使用方法借用代数拓扑，以及现代技术的代数几何。 拓扑学的使用涉及到连续变化的结构族的研究，这些结构族传统上被认为是通过其他手段。 希望这些新技术将提供深入和长期存在的代数几何问题的见解。 代数拓扑学是通过代数技术来研究几何对象。 令人兴奋的新发展导致了群论的进步，使用代数拓扑，从而扭转了通常信息流的方向。 群是发生在所有科学中的基本对称性，包括涉及代码的领域，以及物理学中的结构。 Priddy希望这种新方法能更好地理解这些领域之间的关系。 ***