The Principle of Functoriality

Following the explicit instructions of the organizers, I have tried to write an article that is suitable for a general mathematical audience. It contains some analogies and metaphors that might even be put to nonmathematicians. I hope that experts will be tolerant of the inevitable simplifications. The principle of functoriality is one of the central questions of present day mathematics. It is a far reaching, but quite precise, conjecture of Langlands that relates fundamental arithmetic information with equally fundamental analytic information. The arithmetic information arises from the solutions of algebraic equations. It includes data that classify algebraic number fields, and more general algebraic varieties. The analytic information arises from spectra of differential equations and group representations. It includes data that classify irreducible representations of reductive groups.