Mathematical Sciences: The Structure of Polynomials in Several Variables and Hilbert Identities

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Polynomials in several variables are important and ubiquitous. The principal investigator will work on the creation of new Hilbert Identities; the further analysis of spherical designs; the development of a theory of blenders; the representation of forms as sums of powers of linear forms; and the continuing explorations of sums of powers of rational functions over fields. These studies touch many areas of mathematics including algebra, number theory, combinatorics, algebric geometry, harmonic analysis and functional analysis. 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.