Scanlon investigates the structure of definable sets in theories of enriched fields with an eye towards applications in diophantine and algebraic geometry. In many theories of enriched fields, for example differential, difference and valued D-fields, the definable sets are known to enjoy strong regularity properties, but in most cases the determination of the full induced structure on even one-dimensional sets has proved elusive. Scanlon seeks to determine this structure on minimal sets in differentially closed and difference closed fields and to resolve foundational issues for the theory of valued D-fields in preparation to study the fine structure of their definable sets. The main trichotomy theorem on Zariski geometries, or more exactly the locally modular versus non-orthogonal to an infinite field dichotomy for Zariski groups, implies finiteness and uniformity theorems for certain subgroups and subvarieties of algebraic groups. Scanlon exploits this connection, especially by studying extensions of the Drinfeld module analogue of the Manin-Mumford conjecture to higher dimensional T-modules and of the Tate-Voloch conjecture in the case of p-adically transcendental moduli. Scanlon searches for mathematically meaningful interpretations of the triviality of certain definable sets in differential or difference fields by concentrating on certain subsets of moduli spaces of Shimura varieties definable in some difference closed fields. Scanlon studies inverse problems in model theoretic algebra. Specifically, Scanlon addresses the questions of which fields are stable and which fields are supersimple.
Scanlon undertakes this project to exhibit how the deep, though apparently abstract, theorems of stability theory manifest themselves in concrete mathematical practice. Some such phenomena have already been discovered. Given the strength of these abstract theorems, a systematic search to interpret them can only lead to strong uniformity results on the number and properties of solutions to algebraic, differential, and difference equations not readily perceived from an elementary perspective.
