Cohomological Invariants and Motives of Classifying Spaces

代数几何是研究形状，称为代数簇，由多项式方程定义。本计画发展代数几何的方法，以研究具有大量对称性的代数簇，并使用不变量理论。通过这些方法得到的结果可能会提供一个新的见解的合理性问题的代数簇，衡量代数对象的复杂性。这个项目的大部分内容涉及代数技术的发展，以更好地理解几何中的某些问题。该项目涵盖了代数的广泛方面，如代数几何，代数群和motivic上同调。某些代数群的分类空间可以对经典对象进行分类，例如单代数、对合代数和二次型。PI将研究代数群的上同调不变量。上同调不变量有两个应用。第一个应用程序是有关的代数对象的分类时，对象被确定为同构的上同调不变量。第二个应用程序中考虑的项目是合理性问题的分类空间的代数群。特别是，一个老问题的合理性分类空间的连通代数群的解决。要研究的问题需要精细的技术，包括代数群的表示理论，motivic上同调，代数循环和陈省身classes.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Algebraic geometry is the study of shapes, called algebraic varieties, defined by polynomial equations. This project develops methods of algebraic geometry for the study of algebraic varieties with a large set of symmetries using the theory of invariants. The results obtained by these methods may provide a new insight into the rationality problem of algebraic varieties that measures the complexity of algebraic objects. Much of this project concerns the development of algebraic techniques to obtain a better understanding of certain problems in geometry.The project covers a wide range of aspects in algebra such as algebraic geometry, algebraic groups and motivic cohomology. The classifying spaces of certain algebraic groups classify classical objects such as simple algebras, algebras with involutions, and quadratic forms. The PI will study cohomological invariants of algebraic groups. There are two applications of cohomological invariants. The first application is related to the classification of algebraic objects when the objects are determined up to isomorphism by their cohomological invariants. The second application considered in the project is the rationality problem for classifying spaces of algebraic groups. In particular, an old problem on the rationality of classifying spaces of connected algebraic groups is addressed. The problems to be studied require elaborate techniques including representation theory of algebraic groups, motivic cohomology, algebraic cycles and Chern classes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.