The PI's work is related to two different kinds of symmetries, one algebraic in naure and the other that is analytic. The first kind is symmetries of the roots of polynomial equations defined over the rationals, and the second kind is that which complex analytic functions called modular forms have. The astonishing fact is that there is a connection between these two different kinds of symmetry. This has been implicit in number theoretic work since the time of Gauss, as epitomised in his law of quadratic reciprocity. It has been made explicitly into a unifying theme of modern number theory as part of the Langlands program. The PI's recent work with J-P. Wintenberger on Serre's conjecture results in concrete progress in this program.
The relationship between these two different kinds of symmetries, of roots of polynomial equations and of modular forms, is a phenomenon that has affected a wide spectrum of mathematics and even some parts of physics like string theory. It is a theme that is very old, still resonant in current mathematical research, and promises to be so for a long time to come.
The research carried out in the project was in the subject of number theory. It resulted in the proof of one of the celebrated conjectures in the subject, Serre's modularity conjecture. This work results in establishing deep connections between algebra, geometry and analysis and more particularly Galois groups, motives and automorphic forms. Other work done during the course of the project uses analysis, namely automorphic forms, to construct number fields with given symmetry properties. This is under the rubric of the inverse Galois problem. The broader impact of this project is that it strengthens the deep and pervaive connections between Galois groups, motives and automorphic forms. These connections are at the heart of number thoery and part of the Langlands program. The Langlands program is an overarching program that keeps windening its circle of influence to physics and beyond. The project resulted in the proof of Serre's modularity conjecture, which has been an open and central problem in the subject since the 1970's. It provided new methods of construction of number fields with given symmetry properties. It opened up new connections between higher regulator conjectures and ramification properties of extensions of the rationals. The work of the PI on these projects was carried out with a number of collaborators including Jean-Pierre Wintenberger, Gordan Savin and Michael Larsen. The work introduced new arithmetic techniques to prove substantial and new cases of the Langalnds program including significant new cases of a conjecture of Emil Artin that goes back to the 1930's.
