Quadratic Forms and Division Algebras

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This award supports the study of problems in the algebraic theory of quadratic forms and the theory of finite-dimensional division algebras. Questions considered involve determining the structure of bilinear spaces of elliptic curves. A central problem is to relate the structure of the Witt ring of an elliptic curve over local and global fields to other arithmetic invariants. A second collection of questions involves studying the Brauer group and the valuation theory of division algebras, in both finite and mixed characteristics. Both subjects are intimately 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.