CAREER: Models for Galois deformations and Applications

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Number theory is the branch of mathematics that studies patterns in the arithmetic of whole numbers. A fundamental question, for example, is how many whole number solutions does a particular system of equations have? This is often a very difficult question and engages with techniques from a wide variety of different areas of mathematics. Galois theory, introduced in the 19th century, studies groups of symmetries of polynomial equations and is central to modern number theory. In the 1970s, Langlands introduced a far-reaching web of conjectures which connects Galois symmetries with other symmetric structures appearing in analysis (modular forms). Building bridges between the arithmetic world (Galois theory) and the complex analytic world following Langlands conjectural framework has been a formidable task and a driving force in modern number theory. This project aims to prove new instances of Langlands’ conjectures. This project will open up and engage a number of new questions and concrete problems that will provide research opportunities for graduate students and postdocs. The project will also provide mentoring and online training opportunities for undergraduate and graduate students.In more detail, a major breakthrough in the Langlands program was the proof of Fermat’s Last Theorem which was accomplished by proving a two-dimensional conjecture connecting arithmetic objects (elliptic curves) to analytic objects (modular forms). A key idea in the proof of Fermat’s Last Theorem and in many other important results is the study of congruences modulo a prime p between modular forms. This proposal will systematically construct congruences in higher dimensions by connecting them to new geometric structures in Galois theory. This circle of ideas led to the resolution of Serre’s conjecture for modular forms in the early 2000s, another major achievement in the progression of the Langlands program. The weight part of Serre’s conjecture, which classifies congruences between modular forms, has been generalized in a number of directions as part of the growing field of the p-adic Langlands program. This project will resolve major gaps in progress on these generalizations including the lack of a conjecture for wildly ramified representations and the branching problem for potentially crystalline deformation rings in dimension greater than three. The project will also prove new modularity lifting theorems by studying the geometry of moduli spaces of Galois representations in intermediate weight.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.