Mathematical Sciences: Quadratic Forms, Division Algebras and Galois Cohomology

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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.