Rational curves via function field analytic number theory

Polynomial equations are the cornerstone of modern mathematics. When viewed over the integers, they define Diophantine equations, and questions about the solubility and density of integer solutions have been studied for millennia. When viewed over the complex numbers, they define algebraic varieties, whose classification is a key part of algebraic geometry today, but dates back to 19th Century and an Italian school of geometers. For many years it has been known that the geometry of a system of equations can have a strong influence on their arithmetic. However, in recent years, the flow of information has been reversed, with a surprising link discovered between the geometry of rational curves on higher-dimensional algebraic varieties and questions about the density of solutions to Diophantine equations over global fields of positive characteristic. The main aim of the proposal is to drive this connection forward, by refining a Fourier-analytic circle method approach to tackle central problems in algebraic geometry about moduli spaces of curves on suitable varieties.

This grant investigates fundamental questions in number theory and Diophantine geometry, areas of mathematics that study solutions to polynomial equations using whole numbers or fractions. Such equations are easy to write down, but understanding whether they have solutions - and how many - can be extraordinarily difficult. A key feature of this research is that it also advances algebraic geometry, showing how powerful ideas from number theory can be adapted to address geometric problems. The work brings together tools from analytic number theory, algebra, and geometry. A central theme is the study of rational points: solutions to polynomial equations using rational numbers. These solutions live on geometric objects called varieties, which can be curves, surfaces, or higher-dimensional shapes defined by equations. One of the most basic yet challenging questions is to understand when such solutions exist at all, and how frequently they occur when we restrict their size. Several papers supported by the grant make progress on these questions. In "Integral points on cubic surfaces: heuristics and numerics", a new conjecture is proposed that predicts how densely integer solutions should appear on a broad class of cubic surfaces, combining theoretical insight with extensive numerical evidence. Another highlight, "Pairs of commuting integer matrices", studies a natural algebraic question with connections to symmetry and linear algebra, and establishes a near-optimal upper bound for how often pairs of integer matrices commute. The grant also contributes to algebraic geometry by addressing questions about the geometry of highly complex shapes. The paper "Rational curves on complete intersections and the circle method" investigates which simple, predictable curves - such as lines or parabolas - can lie on intricate, high-dimensional objects known as complete intersections. This line of research is taken significantly further in "Rational surfaces on low degree hypersurfaces", which moves beyond curves to study two-dimensional "flat" objects, called rational surfaces, inside complicated spaces. Using advanced methods from analytic number theory, the latter work provides a new type of connectedness result, describing how these surfaces are arranged and related to one another. Results of this kind are very rare, and it goes far beyond what is currently known in the literature using purely geometric techniques. Overall, the grant demonstrates how ideas from different areas of mathematics can be combined to make progress on long-standing and fundamental problems.