Algebraic K-Theory and Algebraic Groups

摘要 亚历山大默库列夫加州大学，洛杉矶98 01646 该项目涉及代数簇的代数K-理论和代数群理论。 Merkurjev教授打算计算具有大自同构群的代数簇的K-群。 作为这个的一个应用，某些K-上同调群，如周群，将被计算。 研究者还将发展代数簇的上同调不变量理论，并计算给定模块中系数的上同调不变量组。 通过研究对象的对称性来研究对象是现代数学中一个强有力的方法。 数学家们已经发展出相当复杂的方法来测量对称性，他们使用一种称为群的数学对象来跟踪它们。通常，这些群可以堆叠成一个对称性越来越复杂的塔。通过观察塔，他们可以获得大量关于基本物体的信息。代数K理论是这种分组方法的一个版本。 Merkurjev教授有新的想法关于如何计算这些塔的某些类型的对象。 他希望利用这些计算来研究代数簇和代数群。 这些是一组方程的解的集合和特定矩阵集合的数学名称。 这个项目的结果将增加我们对数学的许多基本部分的理解。

ABSTRACT Alexander Merkurjev University of California, Los Angeles 98 01646 This project involves algebraic K-theory of algebraic varieties and the theory of algebraic groups. Professor Merkurjev intends to compute the K-groups of algebraic varieties with large automorphism group. As an application of this certain K-cohomology groups, such as Chow groups, will be computed. The investigator will also develop a theory of cohomological invariants of algebraic varieties and compute the group of cohomological invariants with coefficients in a given module. A powerful method in the modern mathematics is to study an object by studying its symmetries. Mathematicians have developed rather sophisticated ways of measuring symmetry, and they use a mathematical object called a group to keep track of them. Very often the groups can be stacked into a tower of increasingly complicated symmetries. By looking at the tower, they can get a great deal of information about the basic object. Algebraic K-theory is a version of this method of grouping. Professor Merkurjev has new ideas about how to compute these towers for certain kinds of objects. He hopes to use these computations to study algebraic varieties and algebraic groups. These are the mathematical names for a collection of solutions to a set of equations and for particular collections of matrices. The results of this project will add to our understanding of many basic pieces of mathematics.