Model Theory of Fields and Diophantine Geometry

Scanlon研究了富域理论中可定义集的结构，着眼于丢番图和代数几何的应用。在许多富场理论中，例如微分场、差分场和值d场，已知可定义集具有很强的正则性，但在大多数情况下，即使是一维集上的完整诱导结构的确定也被证明是难以捉摸的。Scanlon试图在差闭场和差闭场的极小集上确定这种结构，并解决有值d场理论的基础问题，为研究其可定义集的精细结构做准备。Zariski几何的主要三分定理，或者更确切地说，是Zariski群的局部模与非正交无限场二分定理，包含了代数群的某些子群和子变种的有限性和均匀性定理。Scanlon利用了这种联系，特别是通过研究Drinfeld模类比的Manin-Mumford猜想到高维t模的扩展，以及在p-基超越模的情况下的te- voloch猜想的扩展。Scanlon通过关注在差分闭域中可定义的Shimura变的模空间的某些子集，寻找微分或差分域中某些可定义集合的琐碎性的数学意义解释。Scanlon研究模型论代数中的逆问题。具体来说，Scanlon解决了哪些场是稳定的，哪些场是超简单的问题。Scanlon承担这个项目是为了展示稳定性理论的深刻，虽然表面上是抽象的定理如何在具体的数学实践中体现出来。一些这样的现象已经被发现了。考虑到这些抽象定理的强度，系统地搜索来解释它们只能导致代数、微分和差分方程解的数量和性质上的强一致性结果，而这些结果很难从初等的角度来理解。

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.