Moduli of Galois Representations and Applications

Northwestern University, Evanston IL

Investigators

Abstract

One of the oldest branch of mathematics is Number Theory, which studies the properties and patterns of whole numbers, and which underlies important practical applications such as public key cryptography (widely used for secure communication over the internet). Despite its long history, our understanding of this subject remains unsatisfactory. In the 19th century, E. Galois introduced the idea of studying numbers via the equations that they satisfy, whose (discrete) symmetries are captured in the notion of Galois groups. In the 1970s, Langlands proposed a far reaching web of conjectures which relates these Galois groups to a completely different kind of symmetry, namely the (continuous) symmetries seen in vibrations on highly symmetric shapes (automorphic forms). Progress on particular cases of these conjectures has already led to spectacular achievements, such as the proof of Fermat's Last Theorem, the proof of the Sato-Tate conjecture, solutions of many new Diophantine equations, etc. An emerging theme from these developments is that one can fruitfully study Galois groups (and hence whole numbers) by packaging the information they contain into continuous families, and strikingly these spaces are related to ideas from quantum physics. This project aims to further this circle of ideas, which is essential to make further progress in the Langlands program. More specifically, the project studies the structure of the moduli space of p-adic representations of Galois groups of p-adic fields. This is achieved by modelling deformation spaces of Galois representations with p-adic Hodge theory conditions in terms of spaces studied in Geometric representation theory, especially affine Springer fibers. This creates new links between two different highly-developed and active fields of mathematics. As a result, on one hand, geometric techniques can be used to make substantial progress on major problems in the mod p Langlands program such as Serre weight conjectures, the Breuil-Mezard conjecture, and local-global compatibility problems; on the other hand, insights and heuristics from Galois theory can be used to discover and study new phenomena in modular representation theory and geometric representation theory. Furthermore, the project opens up many concrete questions, both theoretical and computational, which make excellent research opportunities for undergraduate students. 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.