Classifying the Unclassifiables

TL;DR

This paper explores the fundamental dichotomy in the classification of unitary representations of locally compact groups, discusses Mackey's conjecture, and highlights recent developments and potential extensions of these ideas to other mathematical areas.

Contribution

It provides a survey of Mackey's conjecture on representation classifiability, recent progress, and possible extensions to finite p-groups and quantization.

Findings

Verification of Mackey's conjecture by Glimm.

Evidence for extending classification ideas to finite p-groups.

Discussion of Mackey's influence on quantization theory.

Abstract

In 1955 George Mackey suggested that there is a fundamental dichotomy in the unitary representation theory of locally compact second countable groups. He felt that there cannnot be a reasonable classification theory for the unitary representations of a group G for which the dual is a non-smooth Borel space. Mackey's precise conjecture regarding when this is the case was subsequently verified by Glimm. This approach to "classifiability" can be applied in many other branches of mathematics. Included in this article is a sketch of some of the exciting new developments that have been made in this direction. Evidence is given that there should be extensions of Mackey's ideas to such "finitistic" problems as the classification of the finite p-groups. In a different direction, Mackey's thoughts about quantization are also briefly discussed.