Model Theory and Differential Equations

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

These research projects address problems in the model theory of differential fields and the model theory of the field of real numbers with exponentiation. The model theory of differential fields is a fascinating area requiring a sophisticated mixture of ideas from stability theory, differential algebra and algebraic geometry. The work of Buium and Hrushovski has shown that these ideas have important consequences in Diophantine geometry. In particular, the principal investigator will study the model theoretic behavior of solution sets of families of algebraic differential equations. This line of research on the model theory of the field of real numbers with exponentiation is expected to concentrate on the relationship between global solutions to differential equations at infinity and formal solutions in the field of logarithmic-exponential series. Model theory is a branch of logic that explores mathematical structures such as the real numbers together with their arithmetic operations and ordering relation, analyzing the degree to which the basic rules or axioms for a collection of objects and their operations determine the shape of that collection. The methods and results of model theory are cast in terms of definable sets and functions, where a construction is definable exactly when it can be expressed by first-order logical formulas in the language of the structure. Model theoretic methods for the field of real numbers with exponentiation have been remarkably successful in proving new results on the geometry of exponential varieties and sets defined from them. This work has already found applications in asymptotic analysis, control theory, microlocal analysis, and neural networks. The model theory of differential fields has had significant applications in number theory.