Algebraic Cycles On Splitting Varieties

该项目涵盖了代数的广泛方面，如代数几何，代数群和动机上同。研究者提出利用代数循环几何理论研究代数对象的分裂变化的性质。研究者提出研究的第一个课题是代数对象的标准维数，如代数变量、二次型、中心简单代数等。规范维数用来度量一个代数对象的分裂域的复杂度。特别地，研究者提出了计算射影齐次变量和简单代数群的标准维数。研究者还计划与A. Suslin共同计算Severi-Brauer变体的动机上同调，并将此计算推广到任意符号的属分裂变体的情况。这些变种被用于证明米尔诺猜想和布洛赫-加藤猜想。数学的主要目的是提供一种近似于物理世界的图象。该项目从拓扑学发展方法，研究结构的连续变换，称为拓扑空间和代数几何，涉及几何对象，来自多项式方程的图形和称为代数变异。本项目致力于研究动机上同论的某些基本问题-一个相对较新的和非常迅速发展的代数几何分支。代数的新领域将有望从项目工作中发展出来，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.