Algebraic K-Theory and Algebraic Groups

Alexander Merkurjev加州大学洛杉矶分校98 01646本课题涉及代数变异的代数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变种的k群。作为这一理论的应用，我们将计算某些k -上同调群，如Chow群。研究者还将发展代数变量的上同调不变量理论，并计算给定模块中带系数的上同调不变量群。现代数学中一种强有力的方法是通过研究物体的对称性来研究它。数学家们已经发展出相当复杂的测量对称的方法，他们使用一种叫做群的数学对象来跟踪对称。通常情况下，这些群体可以堆叠成一个越来越复杂的对称塔。通过观察这个塔，他们可以得到关于这个基本物体的大量信息。代数k理论是这种分组方法的一个版本。Merkurjev教授对如何为特定类型的物体计算这些塔有了新的想法。他希望用这些计算来研究代数的变种和代数群。这些是一组方程的解的集合和矩阵的特定集合的数学名称。这个项目的结果将增加我们对许多基础数学的理解。

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.