Algebraic Cycles On Splitting Varieties

该项目涵盖了代数几何、代数群和上同调等代数的各个方面，研究人员提出利用代数圈计量学的理论来研究代数对象分裂的性质。研究者建议研究的第一个主题是代数对象的标准维度，如代数簇、二次型、中心单代数等。标准维度衡量了代数对象的一类分裂域的复杂性。特别地，作者提出了计算射影齐次簇和单代数群的标准维数的方法。研究者还计划与A.Suslin共同计算Severi-Brauer簇的基元上同调，并将这一计算推广到任意符号的普通分裂簇的情况。这些变体被用来证明米尔诺和布洛赫-加藤猜想。数学的主要目标是提供对物理世界图景的近似。本课题从拓扑学出发，发展了一种研究几何对象的连续变换的拓扑学方法，称为拓扑空间和代数几何，这些几何对象来自作图多项式方程和代数簇。本课题致力于研究上同调理论的某些基本问题--代数几何中一个相对较新且发展非常迅速的分支。代数的新领域将有望从该项目的工作中演变出来，这将为指导研究生和初级教师创造新的研究机会，并为研究生课程提供材料。

The project covers a wide range of aspects in algebra such asalgebraic geometry, algebraic groups and motivic cohomology.The investigator proposes to use theory of algebraic cycles ingeometry to study properties of splitting varieties of algebraic objects. The first topic the investigator proposes to study is the canonicaldimension of an algebraic object such as algebraic variety,quadratic form, central simple algebra, etc. The canonical dimensionmeasures complexity of the class of splitting fields of an algebraic object. In particular, the investigator proposes to compute the canonicaldimension of projective homogeneous varieties and simple algebraicgroups. The investigator also plans jointly with A. Suslin to compute the motivic cohomology of Severi-Brauer varieties and generalizethis computation to the case of generic splitting varieties of arbitrarysymbols. These varieties were used in the proof of the Milnor and theBloch-Kato Conjecture.The main objective of mathematics is to provide an approximation to thepicture of the physical world. This project develops methods fromtopology that studies continuous transformations of structures calledtopological spaces and algebraic geometry concerning geometric objectscoming from graphing polynomial equations and called algebraic varieties.This project is devoted to the study of certain fundamental problemsof motivic cohomology theory - a relatively new and very quickly developingbranch of algebraic geometry. The new areas of algebra that will hopefullyevolve from the work on the project will create new research opportunities for mentoring graduate students and junior faculty and provide materialfor graduate courses.