Problems in Motivic Cohomology Theory

Proposer计划研究动机上同调理论中的几个主题。第一个项目是比较动机谱序列的两种构造，第一种由Bloch，Lichtenbaum，Friedlander，Suslin和Levine构造，第二种由Grayson和Suslin构造。由于M.Levine通过切片过滤证明了第一个构造与Voevodsky构造一致，这将证明所有已知的构造谱序列的方法都给出了相同的答案。第二个项目是关于非分裂约化代数群的动机，如GL_n，D和SL_n，D，其中D是域F上的中心除法代数。第三个项目是计算域F上的群H^n-1，n(F)。最后一个项目是关于代数余边理论的两种构造的比较。数学的主要目的是给物理世界提供一幅精确的图画，或者至少是这幅图画的一个适当的近似。从这个角度来看，代数变体是最重要的，首先它们是相对容易理解的，因为它们只是由多项式方程定义的，其次它们通常给出一个相当准确的其他形状的近似，最重要的是，它们确实自然地出现在从理论物理到编码理论的相当多的学科中。这就是为什么代数几何--代数变元理论对数学的发展和应用是如此重要的原因。这个项目致力于研究上同调理论的某些基本问题，上同调理论是代数几何中一个相对较新且发展非常迅速的分支。在这部分数学中，几何学与代数和拓扑学混合在一起，要使用的思想和方法同样来自所有这些方向。

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.