Report on the Trace Formula

We report briefly on the present state of the trace formula and some of its applications. This article is a summary of the two hour presentation/discussion on the trace formula. The proposed topic was very broad. It included a recapitulation of the trace formula, past and present, as well as an outlook for its future. The article will treat these matters in only the most concise terms. I include just two references [A] and [L]. The first of these is a general (and detailed) introduction to the trace formula and related topics. It contains references to just about everything discussed in this article. The second is a review by Langlands of his ideas for possible application of the trace formula to the general principle of functoriality. We shall discuss this topic at the end of the article. 1. Invariant trace formula Let G be a connected reductive algebraic group over a global field F of characteristic 0. Then G(F ) embeds as a discrete subgroup of the locally compact adelic group G(A). We write R for the unitary representation of G(A) on L ( G(F )\G(A)) by right translation. For any function f in the global Hecke algebra H(G) (with respect to a suitable maximal compact subgroup K ⊂ G(A)), the average