Model Theory of Fields and Diophantine Geometry

斯坎伦调查结构的可定义集理论的丰富领域的眼睛对应用丢番图和代数几何。在许多丰富域的理论中，例如微分、差分和赋值D-域，已知可定义集合具有强正则性，但在大多数情况下，即使是一维集合上的完全诱导结构的确定也被证明是难以捉摸的。斯坎伦试图确定这种结构的最小集在差分封闭和差分封闭领域，并解决基础问题的理论价值的D-领域准备研究精细结构的可定义集。Zurski几何的主要二分法定理，或者更确切地说，Zurski群的局部模与非正交于无限域二分法，暗示了代数群的某些子群和子簇的有限性和一致性定理。斯坎伦利用这种联系，特别是通过研究延拓德林费尔德模模拟的马宁-芒福德猜想，以高维T-模和泰特-沃洛赫猜想的情况下，p-adically超越模。斯坎伦搜索数学上有意义的解释平凡的某些定义集的微分或差异领域集中在某些子集的模空间的志村品种定义在一些不同的封闭领域。Scanlon研究模型论代数中的逆问题。具体来说，斯坎伦解决了哪些领域是稳定的，哪些领域是超简单的问题。 斯坎伦承担这个项目展示如何深刻的，虽然显然是抽象的，定理的稳定性理论体现在具体的数学实践。 一些这样的现象已经被发现了。 鉴于这些抽象定理的强度，系统的搜索来解释它们只能导致代数，微分和差分方程的解的数量和性质的强一致性结果，从初等的角度来看不容易察觉。

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.