Applications of Model Theory to Functional Transcendence

该项目的重点是研究自然出现在几个物理应用中的某些特殊函数以及数学中的重要问题：几何结构的均匀化函数，Painlevé超越和Schwarzian函数。研究人员将使用模型理论，一个分支的数理逻辑，结合其他领域的数学，如几何和代数，回答深远的问题，在功能超越理论有关的存在代数关系之间的特殊功能。在过去的几十年里，在O-极小性的背景下，随着Pila-Wilkie计数定理的工作，人们对函数超越结果的兴趣激增，部分原因是它们与特殊点集的联系。在这个项目中，研究人员将使用一种全新的方法，围绕微分场的模型理论，攻击一些关于自守函数代数性质的主要开放问题。一个主要的目标是建立Ax-Lindemann-Weierstrass和Ax-Schanuel定理与Shimura簇的均匀化函数的导数以及其他几何结构。该项目还将研究微分封闭域中几何平凡强极小集的分类，并旨在表征Painlevé方程和Schwarzian方程的解集的结构。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.