CAREER: Applied Model Theory

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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.