Classifying the Unclassifiables

George Mackey devoted his career to exploring the basic unity of mathematics, and to understanding its relationships to modern physics. He was a major figure in the flowering of group representation theory and related portions of functional analysis that occured in the middle decades of the last century. Sadly, he, along with many of his colleagues who participated in this endeavor, is no longer with us. Borel classifiability was one of Mackey’s most innovative ideas. Although he only formulated it for unitary group representations, his theory can be adapted to a wide range of mathematical problems. Owing to the efforts of descriptive set-theorists, there has been remarkable progress in this field in the last twenty years. I have sketched some of of these developments, and I have suggested some open problems. I conclude with a few remarks about Mackey’s attempts to come to terms with quantum mechanics. For a more complete and balanced view of George Mackey’s career, the reader should turn to Varadarajan’s masterful article. A number of individuals have made very helpful suggestions for improving earlier versions of this paper. I am particularly indebted to Aleko Kechris, Richard Kadison, and Marc Rieffel.