Algebraic Model Theory

斯坎伦将通过数学逻辑的优势研究差分和微分方程理论，并将这些研究结果应用于算术和几何的基本问题。更具体地说，他将开发一种统一的方法来研究差分和微分场的逻辑理论，并通过射流和弧空间的线性化的一般理论来解决关于差分-微分方程的深层结构问题，从而精确地认识到非线性方程的所有几何复杂性实际上反映了相关线性方程的复杂性。斯坎伦将研究伽罗瓦理论，或对称理论，一般方程组的涉及算子的理论，通过使用这种线性化过程，联络群的逻辑理论，和坦纳基形式主义。此外，他还将利用差分域中可定义集合的精细结构理论，以及通过极小性理论的值域和实几何的模型理论，来解决特殊点上代数关系的具体问题。最后，Scanlon将发展具有解析结构的Witt向量理论和相对的Frobenius模型理论。斯坎伦将把这一理论作为亚稳态一般理论的验证场，并将其作为一种逻辑上驯服的理论，在这种理论中可以编码算术问题，他相信这一理论可以为一种新的动机积分理论奠定基础，在这种理论中，有限维差分变体扮演代数变体的角色。斯坎伦将发展一个独立于素数p的p射流模型理论，从中可以推导出数论问题的均匀性。模型理论，在数学逻辑的意义上，提供了一个统一的视角来研究问题在不同的数学分支作为一般理论的实例。例如，它可以建立一些奇特的，但具有启发性的理论，其中适用于研究几何和微分方程的技术被转换为研究数字。在相反的方向，它非常一般的方法来理解结构的对称性，更重要的是，当专门研究关于微分和差分方程的具体数学问题时，结构的一部分相对于另一部分的对称性揭示了其他未知的对称性理论。在这个项目中，斯坎伦将从数学逻辑、微分方程理论、动力系统、数论等方面探索数学的基本统一。

Scanlon will study the theory of difference and differential equations through the vantage of mathematical logic and to apply the results of these investigations to fundamental problems in arithmetic and geometry.More specifically, he will develop a unified approach to the logical theory of difference and differential fields and to resolve deep structural questions about difference-differential equations through a general theory of linearization via jet and arc spaces giving a precise sense to which all geometric complexity of nonlinear equations actually reflects the complexity of associated linear equations. Scanlon will study Galois theory, or the theory of symmetries, of general systems of equations involving operators by employing this process of linearization, the logical theory of liaison groups, and a Tannakian formalism. In addition, he will address specific questions about algebraic relations on special points by using the fine structure theory for definable sets in difference fields as well as the model theory of valued fields and of real geometry via the theory of o-minimality. Finally, Scanlon will develop the model theory of the theory of the Witt vectors with analytic structure and the relative Frobenius. Scanlon will use this theory both as a proving ground for the general theory of metastability and as a logically tame theory in which arithmetic problems may be encoded believing that this theory may ground a new theory of motivic integration in which finite dimensional difference varieties play the role of algebraic varieties. Scanlon will develop a model theory of p-jets independent of the prime p from which uniformities in number theoretic problems may be deduced.Model theory, in the sense of mathematical logic, gives a unifying perspective for studying questions in disparate branches of mathematics as instances of a general theory. For example, it may ground fanciful, but suggestive, theories in which techniques appropriate to the study of geometry and differential equations are transposed to investigate numbers. In the opposite direction, its very general methods for understanding symmetries of structures and more importantly symmetries of one part of a structure relative to another part when specialized to concrete mathematical questions about differential and difference equations reveal otherwise unknown theories of symmetries. With this project, Scanlon will explore the underlying unity of mathematics as seen from mathematical logic from the theories of differential equations, to dynamical systems, to number theory.