Model theory, diophantine geometry and asymptotic analysis

Purdue University, West Lafayette IN

Investigators

Abstract

The central goal of this project is to make significant progress on solving a diverse array of mathematical problems by developing and exploiting foundational tools and perspectives from model theory, a branch of mathematical logic. An important aspect of model theory is to study the underlying structure of mathematical objects from a formal viewpoint, and then to utilize this to obtain novel approaches to questions emerging from other areas of mathematics, such as algebra, geometry, arithmetic, differential equations, graph theory, or even beyond, such as from economics, machine learning or engineering. The particular focus of this project is on developing recent applications of certain model-theoretic tools to areas such as diophantine geometry (the study of solving equations in integers by means of algebra and geometry) and asymptotic analysis (the study of the growth behaviour of functions as their inputs become large), in order to obtain new insights in these different fields by exploring and profiting from the interplay between them, while also making fundamental contributions to our understanding of the foundational model-theoretic framework itself. The project includes the training of undergraduate and graduate students, including those from underrepresented groups through the Math Alliance. The prevalent feature of the problems that this project proposes to address is their connection to the part of model theory known as o-minimality, a common generalization of semi-algebraic and subanalytic geometries, where the mathematical objects under consideration do not exhibit wild geometric behaviour (such as no oscillation, finiteness bounds being uniform in families, and good asymptotic growth behaviour of functions). A principal focus of this project is on the counting theorem of Pila and Wilkie, a seminal application of o-minimality to diophantine geometry, as well as on subsequent refinements and applications of this result. One key aim is to gain new insights into one of the main analytic-geometric tools used in this area, namely smooth parameterization. Another is to obtain further improvements to or new applications of these counting results, in particular by exploiting effective methods. This project moreover will use techniques from o-minimality to address questions in diophantine and asymptotic analysis, by furthering our understanding of the asymptotic growth behaviour of functions in the o-minimal setting, and making critical new insights into the wider applicability of this analysis. 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.