Motives and Algebraic Groups

动机和代数群

• 批准号: 0098111
• 负责人: Alexander Merkurjev
• 金额: $ 9.67万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-15 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0098111&HistoricalAwards=false
• 关键词: Motives; Algebraic; Groups

The PI of this project will continue to work on problems in algebraic group theory using three algebraic cohomology theories having topological origin: motivic cohomology, algebraic K-theory and algebraic cobordism. The PI proposes to work on three relatively independent topics. The first topic is devoted to the rationality problem of algebraic groups and deals with motivic cohomology. The PI expects to find a Postnikov tower for the motive of a simply connected group involving motives of projective homogeneous varieties. The second topic is related to algebraic cobordism. The PI sees the opportunity to use this theory in order to approach the general Rost's degree formula. The latter has many applications in the theory of homogeneous spaces of algebraic groups. The third topic, the essential dimension, although seemingly different from the others, nevertheless, involves the phenomenon of compression of algebraic varieties and hence is closely related to the second topic. The PI proposes to use degree formulas for the computation of essential dimensions of algebraic groups.The area of this project lies between algebraic geometry, the branch of mathematics devoted to geometric objects called algebraic varieties and described by polynomial equations, and algebraic topology where one studies continuously varying families of structures called topological spaces. Translating the methods of topology from topological spaces to algebraic varieties gives new tools to solve problems in algebraic geometry. Much of this project is about using techniques that are of a topological nature to obtain a better understanding of certain problems in algebraic geometry.

该项目的PI将继续使用具有拓扑来源的三种代数共同体理论来解决代数群体理论中的问题：动机共同体学，代数K理论和代数核心主义。 PI建议研究三个相对独立的主题。第一个主题致力于代数群体的合理性问题，并处理动机共同体。 PI希望找到一座尼科夫塔，以使一个简单相互联系的群体的动机涉及投射均匀品种的动机。 第二个主题与代数恢复有关。 PI看到有机会使用该理论来达到一般的ROST学位公式。 后者在代数群体的同质空间理论中有许多应用。 第三个主题是，基本维度，尽管看似与其他主题有所不同，但仍涉及代数品种压缩的现象，因此与第二个主题密切相关。 The PI proposes to use degree formulas for the computation of essential dimensions of algebraic groups.The area of​​ this project lies between algebraic geometry, the branch of mathematics devoted to geometric objects called algebraic varieties and described by polynomial equations, and algebraic topology where one studies continuously varying families of structures called topological spaces. 将拓扑的方法从拓扑空间转换为代数品种提供了解决代数几何学问题的新工具。 该项目的大部分是关于使用具有拓扑性质的技术，以更好地了解代数几何学中的某些问题。