 
Summary: Contemporary Mathematics
Report on the Trace Formula
James Arthur
This paper is dedicated to Steve Gelbart on the occasion of his sixtieth birthday.
Abstract. 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,
J. Arthur, An introduction to the trace formula, in Harmonic Analysis, the Trace
Formula and Shimura Varieties, Clay Mathematics Proceedings, Volume 4, 2005,
American Mathematical Society, p. 1263.
and
R. Langlands, Un nouveau point de rep`ere dans la th´eorie des formes automorphes,
to appear in Canad. Math. Bull.
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
