Title: P-derivations, Jet Operators and Ring Schemes.

Abstract: One of the remarkable results in Professor A. Buium’s theory of arithmetic differential equations is the classification of jet operators over p-rings. The classification implies the inevitability of p-derivations in any setting involving Z, Z_(p), or Z_p (the integers, their localization at p, and their extension to the p-adic integers). It turns out that this is equivalent to a classification of certain ring schemes. The goal of this talk will be to review some elementary concepts from the theories of arithmetic differential equations and algebraic groups in order to state and prove the classification theorem. We will then look at how and why one might be interested in extending this classification to higher dimensions.