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中的问题研究。考虑的问题涉及确定结构的双线性空间（更确切地说是Knebusch-Witt环）的代数曲线，特别是椭圆曲线。第二组问题涉及研究非驯服的估值理论的司代数，在有限和混合的特点。这两个主题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。支持的研究涉及二次型理论。最简单的形式是研究二次多项式的形式。等价地，它是一个分析类型的内积，可以定义度量几何的n维向量空间。二次型的研究与代数几何和代数k-理论有着深刻的相互关系。

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.