Quadratic Forms and Division Algebras

该奖项支持二次型代数理论和有限维除法代数理论问题的研究。所考虑的问题包括确定椭圆曲线双线性空间的结构。一个中心问题是将椭圆曲线在局部和全局域上的Witt环的结构与其他算术不变量联系起来。第二个问题集涉及研究Brauer群和除法代数的估值理论，在有限和混合特征。这两门学科都与伽罗瓦上同调密切相关，因此伽罗瓦理论中的问题将被视为研究的一部分。所支持的研究涉及二次型理论。它最简单的形式，是对二阶多项式形式的研究。等价地，它是对可以定义n维向量空间度量几何的内积类型的分析。二次型的研究与代数几何和代数k理论有着密切的联系。

This award supports the study of problems in the algebraic theory of quadratic forms and the theory of finite-dimensional division algebras. Questions considered involve determining the structure of bilinear spaces of elliptic curves. A central problem is to relate the structure of the Witt ring of an elliptic curve over local and global fields to other arithmetic invariants. A second collection of questions involves studying the Brauer group and the valuation theory of division algebras, in both finite and mixed characteristics. Both subjects are intimately related to Galois cohomology, and therefore problems in Galois theory will be considered as part of the research. The research supported involves the theory of quadratic forms. This, in its simplest form, is the study of polynomial forms of degree two. Equivalently, it is an analysis of the types of inner products that can define the metric geometry of an n-dimensional vector space. The study of quadratic forms has deep interrelations with algebraic geometry and algebraic K-theory.