Applications of Model Theory to Functional Transcendence

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This project focuses on the study of certain special functions which appear naturally in several physical applications as well as in important problems in mathematics: the uniformization functions of geometric structures, the Painlevé transcendents and the Schwarzian functions. The investigator will use model theory, a branch of mathematical logic, combined with other areas of mathematics, such as geometry and algebra, to answer far-reaching questions in functional transcendence theory related to the existence of algebraic relations among the special functions. Applications of this study to other areas of mathematics, such as number theory, is also a major goal of the project.Over the past decades, following works around the Pila-Wilkie counting theorem in the context of o-minimality, there has been a surge in interest around functional transcendence results, in part due to their connection with special points conjectures. In this project, the investigator will use an entirely new approach, centered around the model theory of differential fields, to attack some of the main open questions about the algebraic nature of automorphic functions. A major goal is to establish the Ax-Lindemann-Weierstrass and Ax-Schanuel Theorems with derivatives for uniformization functions of Shimura varieties as well as other geometric structures. The project will also investigate the classification of geometrically trivial strongly minimal sets in differentially closed fields and aim to characterize the structure of the sets of solutions of the Painlevé equations and the Schwarzian equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.