Modularity and Irrationality

This project aims to shed new light on some of the most fundamental constants in mathematics, such as pi, by uncovering their hidden arithmetic structure. In a major breakthrough, the principal investigator and collaborators recently proved that a certain Dirichlet L-value—a class of numbers that generalizes pi—is irrational, marking only the second such result since the 19th century. These numbers, known as periods, arise from definite integrals and appear throughout mathematics, from number theory to geometry and physics. By advancing the understanding of their irrationality, this research targets one of the deepest and most enduring mysteries in mathematics: what kinds of numbers naturally arise from geometry, and how well can they be approximated by rational numbers? At the same time, the project seeks to illuminate profound connections between geometry and analysis, by deepening the understanding of the Langlands program and its vision of a unified mathematical landscape.

The project pursues two major directions. The first develops new methods to study the irrationality of periods, with the goal of proving landmark results such as the irrationality of Catalan’s constant and improving bounds on how closely pi can be approximated by rational numbers—mirroring famous theorems about algebraic numbers. The second builds on the principal investigator’s recent modularity theorem for a positive proportion of genus 2 curves over the rational numbers—the first of its kind—and aims to extend these results to all such curves, advancing a long-standing frontier of the Langlands program. Together, these efforts aim not only to resolve classical open problems, but also to bring new conceptual clarity to the structure of mathematics itself.

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.

Award Number: 2450123
Principal Investigator: Francesco Calegari
Funds Obligated: $205,000
State: IL