Local and global geometry of moduli spaces of p-adic Galois representations

ongoing

The Langlands correspondence is a series of mathematical predictions describing a deep relationship between two fundamental branches of mathematics: discrete mathematics, which is the basis of counting and number theory, and continuous mathematics, which underlies analysis. These connections are incredibly powerful and have helped solve some of the most challenging problems in mathematics, such as Fermat's Last Theorem, which remained unsolved for nearly 400 years. Despite significant advances over the past 30 years, much of this area remains unexplored. Recently, a new approach has emerged with the potential to revolutionise our understanding of the Langlands correspondence. The expectation is that this correspondence can be achieved via an interpretation of the analytic side as functions on geometric spaces which are built from basic objects on the discrete side—specifically symmetries of solutions to polynomial equations like Y²=X³+X+1. While it is speculated that this viewpoint will substantially simplify the problem, it is presently unclear how exactly these ideas should be implemented. A major issue is that the underlying geometry of these spaces are hard to control, rendering a precise interpretation as functions difficult. My research has hinted that, through innovative combinations of techniques from algebraic geometry and representation theory, taming this geometry is within reach. In this project I will develop these ideas to produce a general framework which controls these spaces. Consequently, foundational results in the Langlands correspondence, currently limited to dimension 2, will be developed in higher dimensions, making inroads into one of the most important and enduring problems of modern number theory.