Modular representation theory, Hilbert modular forms and the geometric Breuil-Mézard conjecture.

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The Langlands program is one of the most striking programs in mathematical research today. It suggests deep connections between algebra, geometry, number theory and analysis. One of the biggest results in this program to date is the proof of Fermat's Last Theorem by Sir Andrew Wiles, which sparked headlines in the national newspapers. The result is as follows: let n be an integer greater than 2, then there are no non-zero integers x, y and z such that x^n + y^n = z^n. While this is a relatively accessible statement, the proof of this result came more than three centuries after it was first conjectured and the efforts towards proving it led to huge innovations in number theory and beyond. The key to proving Fermat's Last Theorem was a deep connection between mathematical objects called modular forms and elliptic curves: the modularity theorem. My work is on a result closely related to this theorem: Serre's modularity conjecture. This was stated by Jean-Pierre Serre in 1973 and refined in 1987, and in fact, has Fermat's Last Theorem, as one of its consequences. The equation in Fermat's Last Theorem is an example of a Diophantine equation, named after the ancient Greek mathematician Diophantus. These are polynomial equations with integer coefficients whose solutions are also restricted to integers. Solving Diophantine equations is one of the main goals of number theory. One way to try to study such equations is to investigate Galois representations, which are mathematical objects capturing symmetries of Diophantine equations. Serre discovered that such representations can be studied by looking at modular forms, which are functions satisfying some nice symmetry properties. This is called the "weak" version of Serre's modularity conjecture. The "strong" version describes the properties of the modular form that corresponds to any given Galois representation. Serre's discoveries were eventually proved correct by Chandrashekhar Khare and Jean-Pierre Wintenberger, building on work of many other mathematicians. The proof relies heavily on earlier work showing that the "weak" version implies the "strong" version. Inspired by work of Kevin Buzzard, Fred Diamond and Frazer Jarvis, I aim to achieve results similar to Serre's modularity conjecture, but in a more general context. This means I work with more complex Galois representations and this causes many intricacies and complications, and this is wherein most of my research lies. I further study other ingredients that also featured in Wiles' proof of Fermat's Last Theorem, in particular geometric objects called "Galois deformation rings". These rings are mathematical objects that carry information about Galois representations. Specifically, they tell you what happens when you take a Galois representation and try to alter it a bit. The geometry of such rings can be described in terms of so-called representation theory: this is a famous result called the Breuil-Mézard conjecture. Matthew Emerton and Toby Gee discovered it was possible to give a more precise description of this geometry of these rings. My work so far has the potential to further enhance Emerton and Gee's work.Building on my work to date, my proposal has three primary goals: (A) to make new advances in modular representation theory,(B) to prove a weight version of "weak" Serre implies "strong" Serre inspired by work of Fred Diamond and Shu Sasaki,(C) apply my work to refine the geometric Breuil-Mézard conjecture.