Algebraic Cycles On Splitting Varieties

该项目涵盖了代数学的广泛领域，如代数几何、代数群和motivic上同调，研究者提出利用代数圈几何学的理论来研究代数对象的分裂簇的性质。研究者提出要研究的第一个主题是代数对象的规范维数，如代数簇，二次型，中心单代数等。规范维数度量代数对象的分裂域类的复杂性。特别地，研究者提出了计算射影齐次簇和简单代数群的标准维数。研究者还与A. Suslin等人计算了Severi-Brauer簇的motivic上同调，并将此计算推广到任意符号的一般分裂簇的情形。数学的主要目的是提供一个近似的物理世界的图片。这个项目从拓扑学中发展方法，研究称为拓扑空间的结构的连续变换和代数几何，涉及来自图形多项式方程的几何对象，称为代数簇。这个项目致力于研究动机上同调理论的某些基本问题-代数几何的一个相对较新和发展很快的分支。代数的新领域将从该项目的工作中迅速发展，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.