LSU
Mathematics

# Overview of Harmonic Analysis and Representation Theory

Harmonic Analysis has a long history. In the 19th century it was much concerned with the expansion of functions in terms of series or integrals of trigonometric functions - ie., Fourier series and Fourier integrals. Such expansions were particularly important because of their effectiveness in the solution of differential equations. Efforts to understand the phenomena of convergence versus divergence of Fourier series were a motivating force behind many developments, including Lebesgue integration, and even Cantor's theory of sets.

In the 20th century it became clear that what made cosines and sines effective in Fourier analysis was the fact that they provide the real and imaginary parts of the homomorphisms of the group of real numbers, under addition, into the group of the complex unit circle, under multiplication. This multiplicative group can be regarded as a group of rotations of the complex plane - ie unitary operators on a one-dimensional `Hilbert' space.

Generally speaking, homomorphisms of a fairly general group into a specific, well-known group are called representations (of the less well understood group) in the other (better understood) group. The main thrust of 20th century harmonic analysis has been to develop harmonic analysis on many non-abelian groups, such as matrix groups or Lie groups, in terms of their representations. But to carry out such a program it was necessary to expand the concept of representation to continuous homomorphisms into the group of unitary operators on infinite-dimensional Hilbert space.

These non-abelian groups are very important in applications as well as in pure mathematics - for example, they are groups of invariants of differential operators or laws of nature. Indeed, when the operators appearing in representations of such groups are expressed coordinate-wise - ie as an infinite matrix - the matrix coefficients turn out to be many of the special functions appearing in classical differential equations. And the operators in representation theory play fundamental roles in quantum mechanics. Some of the earliest 20th century research in harmonic analysis and representation theory, by the mathematical physicist Eugene Wigner, and by mathematicians such as Marshall Stone and John von Neumann, was actually aimed at better understanding of the mathematical foundations of quantum mechanics.

Mathematicians working in Representation Theory often concentrate their efforts in a particular family of groups, since each family has its own idiosyncracies and exhibits different properties. Two of the main classes are semisimple Lie groups, and the related harmonic analysis on symmetric spaces, and nilpotent or solvable Lie groups, and the related harmonic analysis on nilmanifolds or solvmanifolds. Both types of harmonic analysis are areas of research in the LSU Mathematics Department. With its roots deeply embedded in algebra, analysis, and mathematical physics, harmonic analysis and representation theory is an extremely rich subject for investigation, interacting with many parts of both pure and applied mathematics.