CAREER: Applied Model Theory

该项目围绕模型理论的两个广泛应用。第一个与代数和数论有关，第二个是机器学习。模型论是数理逻辑的一部分，在过去的几十年里，它在数学和计算机科学的其他领域得到了广泛的应用。该项目旨在解决与数论和代数微分方程有关的几个长期存在的问题。数论和微分方程是数学的核心领域，在整个科学领域都有应用，本项目旨在解决这些领域的基本问题。在过去的几年里，模型理论和机器学习之间的联系已经出现，这个项目旨在建立在这些联系的基础上，为这两个学科带来新的技术。在过去的几年里，模型理论和机器学习中的开放问题已经使用另一个技术解决了，这个项目将继续下去。机器学习已经成为过去十年中最具影响力的技术之一，并且正在快速增长，该项目旨在使用数理逻辑技术解决机器学习中的基础问题。该项目的教育部分包括通过研究和向高中生推广来培训研究生和本科生。该项目的第一个主要领域是在许多经典背景下出现的非线性常微分方程。该项目将研究自守函数所满足的微分方程，并将结果应用于证明新的超越结果，以及在不太可能的交叉点周围获得丢番图应用。在这里，技术包括分析，微分伽罗瓦理论和几何稳定性理论。将研究Painleve方程，其中心目标是对解之间的代数关系进行分类。这个项目的第二个主要领域是机器学习。在这一领域，模型理论工具将承担解决学习理论中的基础开放问题。例如，将追求在各种环境中的可学习性的私人版本的组合特征。该奖项反映了NSF的法定使命，并被认为值得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来支持。

This project centers around two broad applications of model theory. The first is related to algebra and number theory, and the second is in machine learning. Model theory is a part of mathematical logic which has seen extensive applications in other areas of mathematics and computer science in the last several decades. The project aims to resolve several long-standing problems related to number theory and algebraic differential equations. Number theory and differential equations are central areas of mathematics with applications throughout the sciences, and this project aims to resolve fundamental questions in these areas. Connections between model theory and machine learning have emerged in the last several years, and this project seeks to build on those connections to bring new techniques to both disciplines. In the past several years open problems in both model theory and machine learning have been resolved using techniques from the other, a process this project will continue. Machine learning has emerged as one of the most influential technologies of the last decade and is in the process of rapid growth, and this project aims to attack foundational problems in machine learning using techniques from mathematical logic. The education component of the project involves training of graduate students and undergraduate students through research and outreach to high school students.The first main area of this project centers around nonlinear ordinary differential equations which arise in a number of classical contexts. The project will study the differential equations satisfied by automorphic functions and apply the results to prove new transcendence results as well as obtaining diophantine applications around unlikely intersections. Here, the techniques include analysis, differential Galois theory, and geometric stability theory. Painleve equations will be studied, where the central goal is a classification of the algebraic relations between solutions. The second main area of this project is machine learning. In this area, model theoretic tools will be brought to bear to solve foundational open problems in learning theory. For instance, combinatorial characterizations of private versions of learnability in various settings will be pursued. Learnability in automata and other specific mathematical structures will also be pursued.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.