Summary: TRANSFER, THE FUNDAMENTAL LEMMA, AND THE WORK OF
Automorphic forms are eigenfunctions of natural operators attached to reductive alge-
braic groups. Their eigenvalues are of great arithmetic significance. In fact, the information
they contain is believed to represent a unifying force for large parts of number theory and
arithmetic algebraic geometry.
The Langlands program is a collection of interlocking conjectures and theorems that
govern the theory of automorphic forms. It explains in precise terms how this theory, with
roots in harmonic analysis on algebraic groups, characterizes some of the deepest objects
of arithmetic. There has been substantial progress in the Langlands program since its
origins in a letter from Langlands to Weil in 1967. In particular, it has had applications to
famous problems in number theory, including Artin's conjecture on L-functions, Fermat's
Last Theorem, the Sato-Tate conjecture, and the behaviour of Hasse-Weil zeta functions.
However, its deepest parts remain elusive.
At the center of the Langlands program is the principle of functoriality, a series of
conjectural reciprocity laws among automorphic forms on different groups. There appears
to be no direct way to prove it in any but the simplest of cases. One strategy for more
general cases has been to compare trace formulas. The general trace formula for a reductive
algebraic group G over a number field F is a complex identity, which relates spectral and