Motives and Algebraic Groups

该项目的PI将继续使用具有拓扑起源的三种代数上同调理论：动机上同调、代数k理论和代数上同调来研究代数群论中的问题。PI建议研究三个相对独立的课题。第一个主题是讨论代数群的合理性问题和动机上同。PI期望找到一个波斯特尼科夫塔的动机的单连通群涉及射影齐次变量的动机。第二个主题是关于代数协数。PI看到了利用这一理论来接近一般罗斯特度公式的机会。后者在代数群的齐次空间理论中有许多应用。第三个主题，即基本维度，虽然看起来与其他主题不同，但它涉及代数变量的压缩现象，因此与第二个主题密切相关。PI提出用度公式来计算代数群的基本维数。这个项目的领域介于代数几何和代数拓扑之间，代数几何是数学的一个分支，专门研究被称为代数变量的几何对象，并由多项式方程描述，代数拓扑是研究被称为拓扑空间的连续变化的结构族。将拓扑方法从拓扑空间转化为代数变量，为求解代数几何问题提供了新的工具。这个项目的大部分内容是关于使用拓扑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.