Model Theory and Differential and Difference Equations

在这个项目中，研究者将使用数学逻辑的一个分支模型理论对一些经典的微分和差分方程进行分类和研究。模型论是数学逻辑的一个分支，研究数学结构及其可定义集（即“存在”于这些结构中的基本集）。另一方面，微分方程和差分方程是描述事物如何随时间变化的方程，是现代科学和工程用来模拟物理现实的基本工具。该项目的基本思想是将微分或差分方程的解集视为适当结构中的可定义集，并使用模型理论的工具来理解其性质。该项目关注的方程——painlev<s:1>微分方程和差分方程以及Schwarzian微分方程——出现在许多重要的物理应用中，如统计力学和广义相对论，以及数论中的重要问题。该项目建立在模型理论家最近的成功基础上，利用几何稳定性理论的技术，根据特征为零的微分封闭场的三分定理，对这些微分方程的特殊情况进行分类。研究者将把他在一般的painlev<s:1>（微分）方程上的工作扩展到非一般的方程以及Schwarzian方程。目的是充分描述解集的结构，并利用这种描述来更好地理解理论中的几何平凡集。这项工作有望应用于数论和泛函超越理论中的问题。该项目还将把一些用于研究经典微分方程的思想和技术应用于差分设置。主要的挑战将是发展差分方程的不可约性的概念（在经典函数的意义上），并证明差分painlevel方程是不可约的。除了模型理论，这项工作还将采用代数和几何以及微分/差分方程分析研究中的技术。

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 Painlevé 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 Painlevé (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 Painlevé 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.