Classifying unitary representations

经典傅立叶分析的基本概念是，基本上任意的函数可以表示为指数的线性组合。其哲学是将一个函数空间分解成我们想要回答的问题的子空间(例如。解微分方程式)很容易处理。这一哲学的一个简单例子是对角化矩阵：在特征空间上很容易理解向量乘以矩阵的效果。20世纪30年代，盖尔芬德在抽象调和分析中引入了一个广泛的程序，它推广了这一哲学，允许将数学中不同领域的困难问题，如分析和拓扑学，转移到更容易处理的代数问题上。Gelfand程序的一个未解决的部分是对群的不可约酉表示的分类，称为酉对偶问题。酉对偶问题的解在数学的许多领域都有应用，例如自同构形式、类场理论和数论，也在物理(特别是量子力学)中应用。

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).