Mathematical Sciences: Quadratic Forms and Related Topics

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This project is concerned with research on Witt rings and quadratic forms. The principal investigator will focus on the computation of Witt kernels for algebraic and transcendental field extensions. He will also continue his work on the theory of semidefinite polynomials and sums of even powers in rational function fields over a real-closed field, with particular attention to quantitative questions on the higher pythagoras numbers of such fields. In another direction, he will carry out further investigations on Ore extensions over division rings, in the hope that this will shed some light on the structure of division rings themselves. 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.