Problems in Motivic Cohomology Theory

动机上同调理论中的问题

• 批准号: 0901852
• 负责人: Andrei Suslin
• 金额: $ 19万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901852&HistoricalAwards=false
• 关键词: Problems; Motivic; Cohomology; Theory

Proposer plans to investigate several topics in motivic cohomology theory. The first project is to compare two constructions of the motivic spectral sequence, the first due to Bloch, Lichtenbaum, Friedlander, Suslin and Levine and the second one due to Grayson and Suslin. Since the first construction was shown by M. Levine to coincide with the Voevodsky construction via the slice filtration this will prove that all the known approaches to the construction of the spectral sequence give the same answer. The second project concerns the motives of non split reductive algebraic groups like GL_n,D and SL_n,D, where D is a central division algebra over a field F. The third project is an attempt to compute the group H^n-1,n(F) for a field F. Finally the last project concerns the comparison between two constructions of the algebraic cobordism theory.The main objective of mathematics is to provide an accurate picture to the physical world or at least an appropriate approximation of that picture. From this point of view algebraic varieties are of principal importance, first they are relatively easy to understand since they are just defined by polynomial equations, next they usually give a rather accurate approximation to other shapes, most importantly they do appear naturally in quite a lot of subjects from theoretical physics to coding theory. That is why algebraic geometry - the theory of algebraic varieties is so important for the development and applications of mathematics. This project is devoted to the study of certain fundamental problems of motivic cohomology theory - a relatively new and very quickly developing branch of algebraic geometry. Geometry is blended with algebra and topology in this part of mathematics, ideas and methods to be used come equally from all these directions.

提议者计划研究动机上同调理论中的几个主题。第一个项目是比较两个结构的motivic频谱序列，第一个由于布洛赫，Lichtenbaum，Friedlander，Suslin和莱文和第二个由于格雷森和Suslin。自从M. Levine通过切片过滤与Voevodsky构造相一致，这将证明所有已知的构造谱序列的方法给出相同的答案。第二个项目是关于域F上的中心除代数GL_n，D和SL_n，D的非分裂约化代数群的动机。第三个项目是试图计算域F的群H^n-1，n（F）。最后，最后一个项目涉及代数配边理论的两种构造之间的比较。数学的主要目标是为物理世界提供一个准确的图像，或者至少是该图像的适当近似。从这个角度来看代数簇是主要的重要性，首先，他们是比较容易理解，因为他们只是定义的多项式方程，其次，他们通常给一个相当准确的近似其他形状，最重要的是，他们自然出现在相当多的学科从理论物理编码理论。这就是为什么代数几何-代数簇的理论对数学的发展和应用如此重要。这个项目致力于研究动机上同调理论的某些基本问题--这是代数几何学中一个相对较新且发展很快的分支。在数学的这一部分中，几何学与代数学和拓扑学相结合，所使用的思想和方法同样来自所有这些方向。