Mathematical Sciences: Quadratic Forms and Ring Theory

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This project is concerned with field arithmetic and the algebraic theory of quadratic forms. The principal investigator has a long-term interest in the structure and classification of positive semi-definite polynomials and will continue working in this area. In relation to this work, the principal investigator will study the Hurwitz Problem on the composition of sums of squares by using the techniques of induced representations. The postdoctoral associate on this award will work on the theory of W-groups and will investigate the application of this theory to problems in quadratic forms and field invariants. The research supported involves the theory of quadratic forms which, in its simplest form, is a study of polynomial forms of degree two or equivalently, an analysis of the types of inner products that can define the metric geometry of an n-dimensional vector space. This is an important and difficult branch of mathematics which has attracted the attention of a number of notable mathematicians.