Mathematical Sciences: Quadratic Forms, Division Algebras and Galois Cohomology

该奖项支持二次型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代数曲线上的双线性空间（更准确地说是Knebusch-Witt环）的结构，特别是椭圆曲线。第二组问题涉及研究有限和混合性质的除法代数的非驯服估值理论。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 research on problems in the theory of quadratic forms and the theory of finite-dimensional division algebras. Questions considered involve determining the structure of bilinear spaces (more precisely the Knebusch-Witt ring) over algebraic curves, in particular elliptic curves. A second set of problems involves studying the non-tame valuation theory of division algebras, in both finite and mixed characteristics. Both subjects are intimatley 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.