CAREER: Geometric and Algebraic Model Theory

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Scanlon's work integrates arithmetic, geometry and logic via amodel-theoretic study of natural mathematical structures. Specifically,Scanlon investigates the theory of compact complex analytic manifolds froma model theoretic point of view and extends this research to relatedgeometric objects. In developing the theory of jets of differentialequations, Scanlon hopes to discern the fine structure of dependence onsolution sets to partial differential equations of infinite differentialdimension and to present a unified treatment of differential Galois theoryfor such equations. Scanlon plans to refine the study of enriched valuedfields and to draw out the Diophantine consequences of these results.Scanlon works, as well, on purely logical issues and will continuedeveloping the theory of thorn-independence with an eye towards eventualapplications.Scanlon's work integrates arithmetic, geometry and logic. Definable sets,namely those sets definable within a fixed formal language, are principalobjects of study in model theory and this special emphasis has beeninstrumental in the success of the model-theoretic approaches to numbertheory, algebra and analytic geometry. Scanlon works to tighten theconnection between logical and geometric methods and to find mathematicalinterpretations of theorems in logic.