Classifying unitary representations

经典傅里叶分析的基本概念是，一个基本上任意的函数可以表示为指数的线性组合。其原理是将函数空间分解为子空间，我们希望回答的问题（例如：解一个微分方程）是易处理的。 这种哲学的一个简单例子是对角化矩阵：向量乘以矩阵的效果在特征空间上很容易理解。1930年代，I.M. Gelfand介绍了一个广泛的计划，在抽象的谐波分析，推广这一理念，使转让困难的问题，在不同领域的数学，如分析和拓扑结构，以更易于处理的问题，代数。一个未解决的组成部分Gelfand的计划是分类的不可约酉表示的一组，被称为酉对偶问题。酉对偶问题的解在许多数学领域都有应用，例如自守形式、类场论和数论，也在物理学（特别是量子力学）中有应用。

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).