Mathematical Sciences: Quadratic Forms and Ring Theory

这个项目是关于域算术和二次型的代数理论。首席研究员对半正定多项式的结构和分类有着长期的兴趣，并将继续在这一领域工作。在这项工作中，首席研究员将使用归纳表示法研究关于平方和组成的Hurwitz问题。该奖项的博士后助理将致力于W-群的理论，并将研究该理论在二次型和场不变量问题中的应用。所支持的研究涉及二次型理论，其最简单的形式是研究二次或等价的二次多项式形式，分析可以定义n维矢量空间的度量几何的内积的类型。这是一个重要而又困难的数学分支，引起了许多著名数学家的注意。

This project is concerned with field arithmetic and the algebraic theory of quadratic forms. The principal investigator has a long-term interest in the structure and classification of positive semi-definite polynomials and will continue working in this area. In relation to this work, the principal investigator will study the Hurwitz Problem on the composition of sums of squares by using the techniques of induced representations. The postdoctoral associate on this award will work on the theory of W-groups and will investigate the application of this theory to problems in quadratic forms and field invariants. The research supported involves the theory of quadratic forms which, in its simplest form, is a study of polynomial forms of degree two or equivalently, an analysis of the types of inner products that can define the metric geometry of an n-dimensional vector space. This is an important and difficult branch of mathematics which has attracted the attention of a number of notable mathematicians.