Integer valued polynomials

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My proposed research is in the general areas of algebra and number theory and studies algebras of polynomials determined by integrality conditions on their values, for example the algebra of polynomials with rational coefficients which take integer values when evaluated at integers. This has been a topic of interest within algebra for over a century which is now proving to have applications in other areas of mathematics such as algebraic topology and non-archimedean analysis. My research concentrates on computational aspects of such algebras, constructing algorithms to determine bases and generating sets and for the evaluation of associated invariants. These frequently require number theoretic arguments and produce interesting number theoretic results. The current goals of this research program are a better understanding of such algebras of polynomials when the underlying ring of coefficients is not commutative (for example division algebras or rings of matrices) and general computational methods for such algebras of polynomials in several variables. Results in either of these directions would have useful applications outside of algebra. It has been known since the 1970's that there is a close connection between such algebras and Hopf algebras of operations in generalized cohomology theories such as K-theory and its variants and that homological calculations for these theories can be cast as problems about integer valued polynomials in several variables. Also in non-archimedean analysis the computation of the capacity can be expressed as the evaluation of a limit of certain invariants of associated to such algebras.