Classifying unitary representations

经典傅里叶分析的基本概念是，本质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我们希望回答的问题的子空间(例如；解微分方程是容易处理的。这种哲学的一个简单的例子是对角化矩阵：向量乘以矩阵对特征空间的影响很容易理解。在20世纪30年代，I.M. Gelfand在抽象调和分析中引入了一个广泛的程序，推广了这一哲学，允许将数学中不同领域的难题（如分析和拓扑）转移到更容易处理的代数问题上。Gelfand方案的一个未解决的部分是群的不可约酉表示的分类，称为酉对偶问题。一元对偶问题的解在数学的许多领域都有应用，如自同构形式、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).