LEAPS-MPS: Non-Archimedean, Arithmetic, and Algebraic Geometry

University Of North Texas, Denton TX

Investigators

Abstract

The study of the arithmetic and geometry of solutions to systems of polynomial equations goes back many centuries to the Greeks. There is a deep interplay between these two disciplines, and the guiding principle is that the geometry determines the arithmetic. This principle rings true when the system of polynomial equations defines a curve (i.e., a one-dimensional object), and in particular the number of rational solutions of a curve is governed by the geometry of the set of solutions when considered with complex entries. When the system defines a higher-dimensional object, the geometry becomes much more complicated and as a result so do the arithmetic properties. Instead of trying to understand the algebraic nature of their geometry, mathematicians have sought to understand their complex analytic geometry, and this approach has led to useful analytic characterizations of these solution sets. The main goal of this project is to study the algebraic aspects of a system of polynomial equations using an analytic approach founded in non-Archimedean geometry. This is a topic at the intersection of various areas of research such as complex geometry, non-Archimedean geometry, algebraic geometry, and number theory. In addition to these research activities, the investigator will organize undergraduate learning seminars on topics in number theory and arithmetic geometry which will aid in recruitment and retention in the sciences of our students, many of whom are underrepresented. Furthermore, the investigator will organize several weekend mini-conferences for students which will feature mathematicians from underrepresented groups with the goals of building community and also fostering future mentorship opportunities. The research of this project aims to answer several questions regarding the geometry and arithmetic of varieties of general type. The major goal of this project is to prove a long-standing conjecture of Demailly and Lang regarding the complexity of curves inside of varieties of general type. The investigator has already used non-Archimedean methods to make significant progress towards this conjecture, and this project will continue this methodology through the development of new techniques which utilize recent advances in non-Archimedean pluripotential theory. The secondary goal is to further develop non-Archimedean characterizations of varieties of (log)-general type, primarily focusing on the cases of surfaces and closed subvarieties of commutative algebraic groups. The third goal is to study integral points on quasi-projective varieties with infinite tame fundamental group using methods from algebraic geometry and Diophantine approximation. 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.