Essential Dimension and Cohomological Invariants of Algebraic Groups

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

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.