Mathematical Sciences: Topological Applications of Algebraic Cycles

数学科学：代数圈的拓扑应用

• 批准号: 9401533
• 负责人: Paulo Lima-Filho
• 金额: $ 6.75万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1998-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401533&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topological; Applications; Algebraic

9401533 Lima-Filho The investigator will continue to explore topological properties of the groups of algebraic cycles on algebraic varieties. He has found this to be a mine of applications to topological and algebraic geometric problems. Algebraic cycles are seen as a topological group functor from the category of algebraic varieties to the category of group objects in the category of CW-complexes. This functor associates principal fibrations to closed inclusions, and their homotopy groups provide a homology theory for algebraic varieties which he wishes to understood more deeply. An intersection theory for arbitrary varieties will be developed, generalizing and simplifying an existing theory for quasiprojective varieties. On the purely topological side, the investigator's research has created equivariant infinite loop spaces using algebraic cycles, and the generalized cohomology theories thus obtained have been related with several other theories. As an outcome of this interplay between new and old theories, he has already answered long-standing conjectures about the total Chern class map and is presently computing several examples where algebraic geometry, group representation theory, and homotopy theory come together to illuminate the coefficient systems involved, the associated transfer maps, and so forth. A fundamental aspect and guiding principle in the investigator's research is its unifying character, under which diverse areas of mathematics come into play and yield multifaceted applications of his algebraic techniques. He thus obtains a highly desirable economy of thought, in the sense that a concise and unified approach to various seemingly disparate questions attains deep results about each in a short period of time. He starts by endowing basic objects in algebraic geometry, the algebraic cycles, with a topology which turns out to have remarkable properties. He then realizes that these cycles provide new invariants not only for alge braic varieties (of fundamental importance to algebraic geometers), but that they also deeply relate with ongoing mainstream research in stable homotopy theory, a major topic in algebraic topology. Other natural structures arise in this setting as he explores constructions involving his algebraic cycles. The study of cycles from an equivariant point of view thus also provides insight into various aspects of the theory of representation of finite groups and group cohomology, enhancing the usefulness of his unified approach. ***

小行星9401533 研究者将继续探索代数簇上代数圈群的拓扑性质。 他发现这是一个地雷的应用拓扑和代数几何问题。 代数圈被看作是从代数簇范畴到CW-复形范畴中的群对象范畴的拓扑群函子。 这函人协会主要纤维化封闭的夹杂物，他们的同伦群提供了一个同源理论的代数品种，他希望更深入地了解。 一个相交理论的任意品种将开发，推广和简化现有的理论quasiprojective品种。 在纯拓扑方面，研究者的研究已经使用代数圈创建了等变无限循环空间，并且由此获得的广义上同调理论与其他几个理论有关。 作为新旧理论之间相互作用的结果，他已经回答了关于全陈省身类映射的长期问题，目前正在计算几个例子，其中代数几何，群表示理论和同伦理论结合在一起，以阐明所涉及的系数系统，相关的转移映射等。 一个基本方面和指导原则，在调查员的研究是其统一的字符，根据不同领域的数学发挥作用，并产生多方面的应用，他的代数技术。 因此，他获得了一个非常可取的经济思想，在这个意义上，一个简洁和统一的方法来解决各种看似不同的问题，在很短的时间内获得了关于每个问题的深刻结果。 他首先赋予基本对象在代数几何，代数周期，与拓扑原来具有显着的性质。 然后，他意识到，这些周期提供了新的不变量，不仅为alge numerous品种（具有根本重要性的代数geometers），但他们也深深地关系到正在进行的主流研究稳定同伦理论，一个主要议题，代数拓扑。 其他自然结构出现在这种情况下，他探讨建设涉及他的代数周期。 研究周期从一个等变的角度来看，因此也提供了深入了解各个方面的理论表示的有限群和群上同调，提高实用性，他的统一办法。 ***