Physics Group Meeting

I will introduce polynomial congruences and indicate why number theorists might care about them. I will then describe reciprocity laws which predict the number of solutions to a polynomial congruence in terms of discrete subgroups of Lie groups. The first such law was proved by Gauss, and Langlands has proposed an enormous generalization. I will give some examples to illustrate how these reciprocity laws provide very powerful tools for number theorists. If I have time I will say something about how Galois theory enters the picture.