Let M be an n × n matrix of real numbers. An eigenvalue λ for M is a number such that there is an n-vector v with

M v = λ v .

We call the vector v, an eigenvector of the matrix M. Eigenvalues and eigenvectors tell us about the shape and the action of a given matrix or linear operator on Euclidean space. They are powerful tools in analytical mathematics, in engineering, and in many other parts of modern technology.

In practice it is not necessarily important to know what the actual values of the eigenvalues are. Instead, we tend to concentrate on (i) the signs of the eigenvalues, (ii) the relative sizes of the eigenvalues, and (iii) whether the eigenvalues are real or complex. The collection of eigenvalues is called the spectrum of the matrix, and the study of eigenvalues is called spectral analysis.

In the case that M is a symmetric matrix with real coefficients, it is easy to show that the matrix is diagonalizable (and the diagonalizing process preserves the collection of eigenvalues) and that those eigenvalues will all be real. This is a fundamental situation that says a lot about the stability of physical systems. We are interested in generalizing this particular situation from finite-dimensional Euclidean space to an infinite-dimensional setting. The first and most natural setting in which to begin such a study would be (separable) Hilbert space.

A model for Hilbert space is the collection Η = {(a_j)} of all sequences of complex numbers with the property that ∑_j |a_j|² < ∞. Two elements (a_j) and (b_j) of Η are added by

(a_j) + (b_j) = (a_j + b_j) .

The dot product (or scalar product) of two such elements is

(a_j) ⋅ (b_j) = ∑_j a_j b_j .

If we let a = (a_j), then we define the norm of a in Η by

||(a_j)|| = [ ∑_j |a_j|² ]^1/2 .

The distance of two elements a = (a_j) and b = (b_j) is

d(a, b) = [ ∑_j |a_j - b_j|² ]^1/2 .

Hilbert space bears a strong resemblance to the familiar finite-dimensional Euclidean space, but compactness and convergence questions are much more subtle in this new setting.

Another familiar example of a Hilbert space, actually equivalent to the first one is the collection of periodic functions f on the interval [0,2π] such that

( ∫₀^{2π |f(x)|² dx )^1/2 < ∞ .

Call this Hilbert space Κ. Two elements of Κ are added just as two functions would be added. The inner product of f and g in Κ is defined to be

⟨ f, g ⟩ = ∫₀^2π f(x) [g(x)]^# dx ,

where the # denotes complex conjugation. The norm of f is

||f|| ≡ ( ∫₀^2π |f(x)|² dx ) ^1/2 ,

and the distance between f and g is given by

d(f, g) = ( ∫₀^2π |f(x) - g(x)|² dx )^1/2 .

The Hilbert space Κ is commonly denoted L²(T), and we continue that custom here (the notation T indicates that we are working on the interval [0,2π] with periodicity imposed---in other words we are working on a circle).

The latter Hilbert space is the proper object of study in Fourier analysis. It is known, and this is an instance of the Peter-Weyl theorem, that any element of L²(T) can be written as a superposition (with suitable coefficients) of the exponentials {e^ijx}. And in fact these exponentials are eigenfunctions of the linear operator

L = - d²/dx² .

By applying L to the function e^ijx, we see that the eigenvalues of L are squares of integers.

In dimension 2, we can do a similar analysis on the sphere. This gives rise to the theory of spherical harmonics (instead of exponentials). It is possible---though rather sophisticated and complicated---to write down eigenvalues and eigenfunctions in this situation as well (in this case the operator L will be the negative of the Laplacian: L = - Δ = - ∑_j ∂²/∂ x². It is notable that the sphere is a differential geometric object with constant curvature 1.

Yet another approach to the 2-dimensional situation is to look at the torus, which is T × T. Again it is possible to calculate eigenvalues and eigenfunction. This time we note that the curvature of the surface is 0.

A fundamental result in classical complex analysis (which is one way to look at two-dimensional Euclidean space) is the uniformization theorem. Let U be a 2-dimensional surface. Then U has a universal covering surface U^* which is simply connected. What can this covering surface U^* be? From the point of view of complex analysis there are only three answers: the Riemann sphere C^*, the plane C, and the unit disc D = {z ∈ C: |z| < 1}. The sphere only occurs for U^* when the original surface U is a sphere. The plane only occurs for U^* when the original surface U is a plane, a cylinder, a torus, or a punctured plane. These are all constant curvature situations. The most fascinating and complex set of surfaces from our point of view are the ones for which U^* = D. These are called hyperbolic, and they have negative curvature.

The hyperbolic surfaces are of great interest, and they are not completely understood even today (after more than 150 years of study). It is no longer possible, in this generality, to calculate explicitly the eigenvalues and eigenfunctions.

There are special hyperbolic surfaces for which the spectral analysis has consequences for number theory. In particular, it is useful for number theorists to study the surfaces that arise by taking the quotient of the upper half plane (which is conformally equivalent to the disc) by subgroups of SL(2, Z). Here SL(2,Z) is the group (under multiplication) of 2 × 2 matrices with integer coefficients and having determinant 1. This particular AIM workshop was interested in those surfaces.

One breakthrough presented at the workshop was that Ce Bian, a student of Professor Andrew Booker of Bristol, England, was able to construct GL(3) automorphic forms using indirect numerical techniques. His work involved 10,000 hours of computer time, and is an important scientific calculation that will have many consequences..