Modern Approaches for Classical Diophantine Problems

This research project concerns Diophantine approximation and the geometry of numbers. Diophantine approximation deals with approximation of real numbers by rational numbers and with questions of classification of given numbers as irrational, algebraic, or transcendental. Diophantine equations are algebraic equations with integer coefficients, for which integer solutions are sought. The geometry of numbers deals with the use of geometric notions to solve problems in number theory, usually via the solutions of equations in integers. One of the basic problems of mathematics is to find the solutions of a given polynomial equation. The level of difficulty of such problems depends on the shape of the polynomial and more so on what kind of solutions one is looking for. Finding complex numbers that are solutions to a polynomial equation is a relatively easy task. However, finding integer solutions to polynomial equations is a problem of much greater subtlety and depth, and is the focus of number theory. This project aims to broaden and deepen knowledge in this fundamental area.

The purpose of this research project is to study the general problem of counting integral solutions of Diophantine equations and its applications. The PI will develop methodological advances by combining techniques from classical analysis with modern applications of analytic number theory and arithmetic geometry to address some important and long standing Diophantine problems effectively and explicitly. The research is also concerned with studying the distribution of integral solutions to Diophantine equations. Results of the project are anticipated to lead to substantially better understanding of the geometric and analytic properties of certain arithmetic objects, such as elliptic curves.