Scanlon's work integrates arithmetic, geometry and logic via a model-theoretic study of natural mathematical structures. Specifically, Scanlon investigates the theory of compact complex analytic manifolds from a model theoretic point of view and extends this research to related geometric objects. In developing the theory of jets of differential equations, Scanlon hopes to discern the fine structure of dependence on solution sets to partial differential equations of infinite differential dimension and to present a unified treatment of differential Galois theory for such equations. Scanlon plans to refine the study of enriched valued fields and to draw out the Diophantine consequences of these results. Scanlon works, as well, on purely logical issues and will continue developing the theory of thorn-independence with an eye towards eventual applications.
Scanlon's work integrates arithmetic, geometry and logic. Definable sets, namely those sets definable within a fixed formal language, are principal objects of study in model theory and this special emphasis has been instrumental in the success of the model-theoretic approaches to number theory, algebra and analytic geometry. Scanlon works to tighten the connection between logical and geometric methods and to find mathematical interpretations of theorems in logic.
