Mathematical Sciences: The Algebraic Theory of Quadratic Forms and Etale Cohomology

这个项目是关于二次型在不同于2的特征域上的代数理论。首席研究员将利用伽罗瓦上同调和各种伽罗瓦模的系数来研究维特环的无扭性质。他还将研究在任意域上定义的椭圆曲线上的二次型。特别地，他将使用ettale上同调来研究这种形式的Witt环，特别是当定义在局部域上时。这个项目是关于二次型理论的。二次型是二阶齐次的多变量多项式函数。非2特征域上的二次型自然是由这些域上向量空间上的对称内积产生的。因此，这些形式与空间的几何形状密切相关。

This project is concerned with the algebraic theory of quadratic forms over fields of characteristic different from two. The principal investigator will investigate torsion-free properties in the Witt ring using Galois cohomology with coefficients in various Galois modules. He will also study quadratic forms over elliptic curves defined over an arbitrary field. In particular, he will use etale cohomology to investigate the Witt ring of such forms, especially when defined over a local field. This project is concerned with the theory of quadratic forms. A quadratic form is a polynomial function of several variables which is homogeneous of degree two. Quadratic forms over fields of characteristic other than two arise naturally from the symmetric inner products on vector spaces over these fields. Thus the forms are intimately connected to the geometry of the space.