Modular Galois representations

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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.