Summary: BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 40, Number 1, Pages 3953
Article electronically published on October 10, 2002
THE PRINCIPLE OF FUNCTORIALITY
Following the explicit instructions of the organizers, I have tried to write an arti-
cle 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 math-
ematics. It is a far reaching, but quite precise, conjecture of Langlands that relates
fundamental arithmetic information with equally fundamental analytic informa-
tion. 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