Applications of Model Theory to Functional Transcendence

本项目主要研究在一些物理应用和重要的数学问题中自然出现的一些特殊函数：几何结构的均匀化函数、painlevel超越和Schwarzian函数。研究者将使用数学逻辑的一个分支模型理论，结合其他数学领域，如几何和代数，来回答与特殊函数之间代数关系存在相关的泛函超越理论中影响深远的问题。将这项研究应用于数学的其他领域，如数论，也是该项目的主要目标。在过去的几十年里，在0极小性背景下围绕Pila-Wilkie计数定理的工作之后，对函数超越结果的兴趣激增，部分原因是它们与特殊点猜想的联系。在这个项目中，研究者将采用一种全新的方法，以微分场的模型理论为中心，来解决一些关于自同构函数的代数性质的主要开放性问题。一个主要的目标是建立Ax-Lindemann-Weierstrass定理和Ax-Schanuel定理及其导数对于Shimura变及其他几何结构的均匀化函数。该项目还将研究微分闭域中几何平凡强极小集的分类，并旨在表征painlev<e:1>方程和Schwarzian方程的解集的结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.