Motives and Algebraic Groups

PI的这个项目将继续工作的问题，代数群理论使用三个代数上同调理论具有拓扑起源：motivic上同调，代数K-理论和代数协边。 PI建议就三个相对独立的主题开展工作。第一个主题是致力于合理性问题的代数群和处理motivic上同调。 PI期望找到一个Postnikov塔的动机，一个单连通群涉及动机的投射齐次品种。 第二个主题与代数配边有关。 PI看到了使用这个理论来接近一般Rost度公式的机会。 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，虽然看起来与其他主题不同，但涉及代数簇的压缩现象，因此与第二个主题密切相关。 PI建议使用度公式计算代数群的基本维数。该项目的领域介于代数几何和代数拓扑之间，代数几何是数学的分支，专门研究称为代数簇的几何对象，并由多项式方程描述，代数拓扑研究称为拓扑空间的连续变化的结构家族。 将拓扑空间的拓扑方法转化为代数簇的拓扑方法，为代数几何问题的求解提供了新的工具。 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.