Classifying unitary representations

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The fundamental concept of classical Fourier analysis is that an essentially arbitrary function may be expressed as a linear combination of exponentials. The philosophy is to decompose a function space into subspaces for which the question we wish to answer (eg. solving a differential equation) is tractable. A simple example of this philosophy is diagonalizing a matrix: the effects of multiplying a vector by a matrix are easy to understand on eigenspaces. In the 1930s, I.M. Gelfand introduced a broad programme in abstract harmonic analysis which generalized this philosophy to permit the transfer of difficult problems in diverse areas of mathematics such as analysis and topology to more tractable problems in algebra. An unresolved component of Gelfand's programme is the classification of the irreducible unitary representations of a group, known as the unitary dual problem. A solution to the unitary dual problem has applications in many areas of mathematics such as automorphic forms, class field theory, and number theory, and also in physics (particularly quantum mechanics).