Motives and Algebraic Groups

这个项目的PI将继续使用三个具有拓扑起源的代数上同调理论来研究代数群论中的问题：Motivic上同调、代数K-理论和代数上同调论。国际和平研究所建议就三个相对独立的主题开展工作。第一个主题致力于代数群的合理性问题，并处理动机上同调。PI期望为一个单连通群的动机找到一个波斯尼科夫塔，该群的动机涉及射影齐次簇的动机。第二个话题是与代数余边法有关的。PI看到了使用这一理论的机会，以便接近一般的罗斯特学位公式。后者在代数群的齐性空间理论中有许多应用。第三个主题，本质维度，虽然看似不同于其他主题，但却涉及代数簇的压缩现象，因此与第二个主题密切相关。PI建议使用度公式来计算代数群的本质维度。这个项目的范围介于代数几何学和代数拓扑学之间，代数几何学是专门研究称为代数簇的几何对象并用多项式方程描述的数学分支，而代数拓扑学研究的是称为拓扑空间的连续变化的结构族。将拓扑学的方法从拓扑空间转化为代数变体，为解决代数几何中的问题提供了新的工具。这个项目的大部分内容是关于使用具有拓扑学性质的技术来更好地理解代数几何中的某些问题。

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.