CAREER: Geometric and Algebraic Model Theory

斯坎伦的工作通过对自然数学结构的模型理论研究，将算术、几何和逻辑结合在一起。具体地说，Scanlon从模型理论的角度研究了紧致复解析流形的理论，并将这一研究扩展到相关的几何对象。在发展微分方程射流理论时，Scanlon希望辨别无穷微分维偏微分方程解依赖于解集的精细结构，并提出这类方程的微分伽罗瓦理论的统一处理方法。Scanlon计划改进对丰富的有价值领域的研究，并得出这些结果的丢番图结果。Scanlon还致力于纯粹的逻辑问题，并将继续发展刺无关理论，着眼于最终的应用。Scanlon的工作融合了算术、几何和逻辑。可定义集，即那些可在固定的形式语言中定义的集合，是模型论的主要研究对象，这种特别的强调对数论、代数和解析几何的模型论方法的成功起到了重要作用。Scanlon致力于加强逻辑方法和几何方法之间的联系，并在逻辑中找到定理的数学解释。

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.