Algebraic Cycles On Splitting Varieties

分裂簇上的代数环

• 批准号: 0652316
• 负责人: Alexander Merkurjev
• 金额: $ 52.75万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652316&HistoricalAwards=false
• 关键词: Algebraic; Cycles; Splitting; Varieties

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.

该项目涵盖了代数几何、代数群、动机上同调等代数领域的多个方面。研究者提出利用几何中的代数圈理论来研究代数对象分裂簇的性质。研究者提出研究的第一个主题是代数对象的规范维数，例如代数簇、二次型、中心简单代数等。规范维数衡量代数对象的分裂域类的复杂性。特别是，研究者建议计算射影齐次簇和简单代数群的规范维数。研究人员还计划与 A. Suslin 联合计算 Severi-Brauer 簇的动机上同调，并将此计算推广到任意符号的通用分裂簇的情况。这些变体被用于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的近似描述。该项目开发了拓扑学方法，研究称为拓扑空间和代数几何的结构的连续变换，涉及来自绘制多项式方程和称为代数簇的几何对象。该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且发展非常迅速的分支。代数的新领域有望从该项目的工作中发展出来，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。