In this project, the investigator will use model theory, a branch of mathematical logic, to classify and study some classical differential and difference equations. Model theory is a branch of mathematical logic that studies mathematical structures and their definable sets (i.e., the basic sets that "live" in those structures). On the other hand, differential and difference equations are equations that describe how things change with time and are the fundamental tools that modern science and engineering use to model physical reality. The basic idea of the project will be to view the set of solutions of a differential or difference equation as a definable set in an appropriate structure, and use the tools of model theory to understand its properties. The equations that the project focuses on -- the PainlevÃ© differential and difference equations and the Schwarzian differential equations -- appear in many important physical applications such as statistical mechanics and general relativity, as well as in important problems in number theory.
The project builds on recent successes by model theorists to use techniques from geometric stability theory to classify special cases of these differential equations according to the trichotomy theorem in differentially closed fields of characteristic zero. The investigator will extend his work on the generic PainlevÃ© (differential) equations to non-generic ones as well as to the Schwarzian equations. The objective is to both fully describe the structure of the sets of solutions and to use such descriptions to better our understanding of geometrically trivial sets in the theory. This work is expected to have applications to problems in number theory and functional transcendence theory. The project will also transport some of the ideas and techniques used to study classical differential equations to the difference setting. The main challenge will be to develop a notion of irreducibility (in the sense of classical functions) for difference equation and show that the difference PainlevÃ© equations are irreducible. In addition to model theory, the work will also employ techniques from algebra and geometry and from the analytic study of differential/difference equations.