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.
