Applications of Model Theory

About This Grant

Model theory is a part of mathematical logic which has extensive applications in other areas of mathematics and computer science. This project centers around three main projects involving model theory. The first involves solutions of differential and difference equations. These types of equations specify how an object or variable moves with respect to another in a continuous or discrete manner, respectively. Over the last decade, model theory has played a pivotal role in the resolution of several long-standing open problems for algebraic differential equations. This project aims to continue that progress as well as adapt the new methods to solutions of difference equations. The second project aims to develop connections between model theory and machine learning on both a theoretical and practical level. The third main area of this project involves applying the lessons learned from machine learning and difference equations in more general model theoretic settings. These adaptations are expected to lead to fundamental new advances in model theory. This project involves graduate student training. Model theory has a long history of applications to transcendence results for differential equations. In the last decade, this circle of results has rapidly expanded as model theoretic methods have become more refined. This project seeks to adapt these results to the setting of difference fields, which is expected to have applications in algebraic dynamics via characterizing the invariant subvarieties of algebraic maps of varieties. In model theory, one usually does not study arbitrary first order theories due to well-known wildness phenomena such as undecidability. Because of this, different settings have developed specific techniques in the presence of various tameness principles. Surprisingly, these principles proved to be the key to efficient learnability in several different models of machine learning. This project seeks to continue to develop this growing area of connection while applying cutting edge results from graph theory and combinatorics. The techniques developed in these projects are expected to have bearing on the development of model theory in the area of simple and stable theories. 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.