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Rouse Ball Lecture

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Counting solutions to congruences: reciprocity laws and density theorems. Reciprocity laws provide a rule to count the number of solutions of a fixed polynomial equation modulo a variable prime number. The rule will involve very different objects: automorphic forms and discrete subgroups of Lie groups. The prototypical example is Gauss’ law of quadratic reciprocity, which concerns a quadratic equation in one variable. Another celebrated example is the Shimura-Taniyama conjecture which concerns a cubic equation in two variables. I will start with Gauss’ law and work my way up to somewhat more complicated examples. At the end of the talk I hope to indicate the current state of our knowledge.

This talk is part of the Faculty of Mathematics Lectures series.

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