CAREER: Galois Representations: Deformation Theory and Motivic Origins

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For centuries, symmetry has played a central organizing role in the study of our mathematical and physical world. What may have begun as simple curiosity--for instance, the impulse to construct and classify the regular polyhedra nearly two and a half thousand years ago--has turned out time and again to be the most effective way to study some of the most fundamental objects of mathematics and physics, whether through the more concrete symmetries of geometrical objects or the more abstract (but essential to our digital world) symmetries of quantum mechanics. Perhaps the most surprising role of symmetry in the sciences is in the study of the prime numbers, the fundamental objects of arithmetic, where there are no manifest "symmetries" such as one encounters in geometry. Nevertheless, much of our deepest knowledge of prime numbers comes from their connections with the "symmetries" of polynomial equations, a subject known as Galois theory. This research project will study Galois representations, which are natural packages for algebraically encoding information about prime numbers. One of the central programs of modern number theory, proposes new ways to "unpack" Galois representations by relating them to remarkably different mathematical objects arising in geometry. The PI will continue his research in this direction to describe these mysterious but absolutely fundamental relationships. As part of the educational component of this CAREER project, the PI will undertake a series of educational projects serving a variety of audiences. He will continue to run an intensive summer number theory program for Utah high school students, introducing them to mathematics as an object of experimentation and discovery, and thereby encouraging them to develop the habits of mind essential to creative intellectual work. This program involves both graduate students and local high school teachers as co-teachers, who can then carry its distinctive pedagogical model with them to other educational settings. With a view toward exciting a broader mathematical public, the PI will also, in conjunction with teaching a history of mathematics course at The University of Utah, develop curricular materials, particularly videos, to be disseminated online, in the history of mathematics. Finally, he will continue his work training PhD students. In more detail, the Langlands program is a series of conjectures that guide much contemporary work in number theory, and in particular provide the deepest conjectural answers to problems relating Galois theory and prime numbers. In doing so, they reach out from number theory to algebraic geometry, representation theory, and beyond. The PI complete two main projects within the very broad purview of the Langlands program. The first concerns the deformation theory of Galois representations, one of the two pillars on which the proof of Fermat's Last Theorem was built, and ever since one of the central research areas within algebraic number theory. Here the PI will study the deformation theory of Galois representations valued in general reductive groups; broadly, this work aims at generalizations of Serre's famous modularity conjecture. The second main project concerns the relationship between Galois representations and motives, the latter being in some sense the best linear approximation to the category of algebraic varieties, of fundamental interest in its own right, but also conjecturally the algebro-geometric counterpart of Galois representations. Here the PI will study a variety of problems concerned with establishing the motivic origin of Galois representations. These include instances of the Fontaine-Mazur conjecture related to the PI's generalized Kuga-Satake theory; study of anabelian properties of moduli spaces; and motivic constructions underlying fundamental objects of geometric representation theory.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.