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

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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.