Applications of Model Theory to Functional Transcendence

该项目重点研究在一些物理应用以及数学中的重要问题中自然出现的某些特殊函数：几何结构的均匀化函数、Painlevé 超越函数和 Schwarzian 函数。研究者将利用数理逻辑的一个分支——模型论，与几何和代数等其他数学领域相结合，来回答与特殊函数之间代数关系的存在有关的函数超越理论中影响深远的问题。将这项研究应用于数论等其他数学领域也是该项目的一个主要目标。在过去的几十年里，随着在 o 极小性背景下围绕 Pila-Wilkie 计数定理的研究，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猜想的联系。在这个项目中，研究人员将使用一种全新的方法，以微分场模型理论为中心，来解决有关自守函数代数性质的一些主要开放问题。主要目标是建立 Ax-Lindemann-Weierstrass 和 Ax-Schanuel 定理及其导数，用于 Shimura 簇的均匀化函数以及其他几何结构。该项目还将研究微分闭域中几何平凡强极小集的分类，旨在表征 Painlevé 方程和 Schwarzian 方程解集的结构。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.