Cohomological Invariants and Motives of Classifying Spaces

代数几何是研究形状的学科，称为代数簇，由多项式方程定义。这个项目发展了代数几何的方法，利用不变量理论研究具有大量对称性的代数簇。通过这些方法得到的结果可以为度量代数对象的复杂性的代数簇的合理性问题提供新的见解。这个项目主要涉及代数技术的发展，以更好地了解几何中的某些问题。该项目涵盖了代数几何、代数群和上同调等代数的广泛方面。某些代数群的分类空间对经典对象进行分类，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。第一个应用涉及代数对象的分类，当对象由其上同调不变量确定为同构时。项目中考虑的第二个应用是代数群空间分类的合理性问题。特别地，讨论了连通代数群空间分类的合理性问题。要研究的问题需要复杂的技术，包括代数群的表示理论、动机上同调、代数圈和陈类。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.