Essential Dimension and Cohomological Invariants of Algebraic Groups

该项目涵盖了代数的广泛方面，如代数群、代数几何和动机上同调。PI提出利用代数几何中的代数圈、代数上边线、代数堆叠和动机上同调理论来研究代数对象的本质维度和代数群的分类空间的合理性性质。这两个主题都通过代数群的上同调不变量的概念联系在一起。PI建议研究的第一个主题是各种代数对象的本质维度。本质维度衡量了一类给定代数对象的复杂性。特别是，PI建议计算某些类代数群的本质维度，并给出在简单代数和二次型理论中的应用。第二个主题是研究代数群的上同调不变量及其在代数群分类空间的稳定合理性这一经典问题上的应用。PI建议研究半单群的极大代数环面的上同调不变量。数学的主要目的是提供对物理世界的图像的近似。这个项目从本质维度出发，发展了研究代数对象的复杂性的方法，以及与几何对象有关的代数几何，这些几何对象来自于绘制称为代数簇的多项式方程。本课题致力于利用代数群不变量的方法研究代数中的某些基本问题。用这种方法得到的结果可以为代数簇的合理性问题中的疑难猜想提供一些有启发性的例子。代数的新领域有望从该项目的工作中发展出来，这将为研究生和初级教师创造新的研究机会，并为研究生课程提供材料。

The project covers a wide range of aspects in algebra such as algebraic groups, algebraic geometry, and motivic cohomology. The PI proposes to use the theory of algebraic cycles, algebraic cobordisms, algebraic stacks and motivic cohomology in algebraic geometry to study the essential dimension of algebraic objects and the rationality property of classifying spaces of algebraic groups. Both topics are related via the notion of cohomological invariants of algebraic groups. The first topic that the PI proposes to investigate is the essential dimension of various algebraic objects. The essential dimension measures the complexity of a given class of algebraic objects. In particular, the PI proposes to compute the essential dimension of certain classes of algebraic groups and give applications in the theory of simple algebras and quadratic forms. The second topic is the study of cohomological invariants of algebraic groups and their applications to the classical problem of stable rationality of classifying spaces of algebraic groups. The PI proposes to study cohomological invariants of maximal algebraic tori of semisimple groups.The main objective of mathematics is to provide an approximation to the picture of the physical world. This project develops methods from the essential dimension that studies the complexity of algebraic objects and algebraic geometry concerning geometric objects coming from graphing polynomial equations that are called algebraic varieties. This project is devoted to the study of certain fundamental problems in algebra using methods of invariants of algebraic groups. Results obtained from this approach may provide enlightening examples related to difficult conjectures in the rationality problem of algebraic varieties. The new areas of algebra that will hopefully evolve from the work on the project will create new research opportunities for graduate students and junior faculty and provide material for graduate courses.