The creators of the Park City Mathematics Institute were brave people. Imagine their proposal… "Let's have yearly multi-week meetings to which we'll bring undergraduate students, graduate students, researchers, high school teachers, and undergraduate faculty; we'll set up a general theme and produce programming at all levels connected somehow to that topic. Oh, yes, and the first year's topic will be Quantum Field Theory." I suspect most of us would have told them they were dreaming.
And yet, here is volume 12 in the IAS/PCMI series collecting lecture notes from the graduate courses offered at PCMI. The Institute has thrived. With the support of the Institute for Advanced Studies, PCMI has become a regular fixture among summer programs in mathematics. It offers mathematicians at all levels a chance to glimpse the incredible variety of things that other people do. The interaction is always interesting and often quite productive.
The topic for this 12th year was Automorphic Forms, which is a notoriously difficult area, basically because getting to the research frontier requires a huge amount of technical background. The book under review collects only the notes for the graduate courses offered at PCMI. (The notes for the undergraduate courses are sometimes published in the AMS's "Student Mathematics Library" series.)
The idea for the graduate courses at PCMI is to take students who already have a good foundation in advanced mathematics and to get them quickly to the point of doing their own research. The articles collected here, therefore, assume quite a bit of the reader. For example, Borel's introductory lectures on "Automorphic Forms on Reductive Groups" pretty much assume that the reader is well-versed in Lie theory and has a good knowledge of the classical theory of modular forms (i.e., the SL(2) case). The other chapters/lectures make similar assumptions.
This book is really for those with the right background, then, and for them it will be of great value. The authors are some of the greatest masters in their fields, and they have made a real effort to open up the subject. Clozel, for example, explains in the foreword to his lectures that he has sought "the simplest formulation" for Arthur's conjectures. The resulting combination of insight and (relative!) readability is what makes this book valuable.
Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME. He is the editor of MAA Reviews
BLL* — The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.