Modular Galois representations

PI的工作与两种不同的对称性有关，一种是自然代数的，另一种是解析的。第一种是定义在有理数上的多项式方程的根的对称性，第二种是称为模形式的复解析函数具有的对称性。令人惊讶的事实是，这两种不同类型的对称性之间存在联系。这一点自高斯时代以来就隐含在数论工作中，他的二次互易定律就是一个缩影。作为朗兰兹计划的一部分，它已明确成为现代数论的统一主题。PI最近与J-P合作的工作。温特伯格关于Serre猜想的结果导致了这个程序的具体进展。这两种不同类型的对称之间的关系，多项式方程的根和模形式，是一个现象，影响了数学的广泛领域，甚至物理的一些部分，如弦理论。这是一个非常古老的主题，在当前的数学研究中仍然引起共鸣，并有望在未来很长一段时间内如此。

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.