A surface is a collection of points organized in a two-dimensional pattern. That is, the points close to a given point are specified by two coordinates, just as points in the plane are given by their x- and y-coordinates. Figure 1 exhibits a 2-dimensional surface. There are many ways of studying geometry on surfaces, as well as studying analysis on surfaces. By using concepts of symmetry, these geometric and analytic structures can profitably be studied using an algebraic construct called a group.

Figure 1

A group is a set G together with a binary operation that is usually denoted by · . We require that the binary operation be associative, that the group have an identity element (usually denoted by e), and that each group element have a multiplicative inverse. For example, the set of all integers with the binary operation of addition is a group. The set of all 2 × 2 matrices with non-vanishing determinant (equipped with ordinary matrix multiplication) is a group.

The fundamental group of a surface, (a "surface group") is a noncommutative group which describes the topology of the surface. More precisely, one forms a group from the set of loops that live in the surface. Figure 2 suggests what the fundamental group of a torus is.

Figure 2

Beginning in the late 19th century through the work of Klein, Poincaré, and Koebe, analytic and geometric structures on surfaces could be interpreted in terms of representations of surface groups in Lie groups. Here a Lie group is a group that is also a geometric object---usually a surface of some kind. Perhaps the simplest nontrivial Lie group is a circle. Obviously a circle is a 1-dimensional surface, but the circle can naturally be identified with the set of rotations in the plane---and that is a group. A torus is also a Lie group for similar reasons. Here a "representation" is a homomorphism of the given group to a more standard object. One learns about the given group by comparison with the standard. In particular, questions concerning integration of algebraic functions could be related both to conformal geometry (the geometry of angles, where considerations of distance are ignored) and to non-Euclidean geometry.

More recently algebraic, analytic and geometry questions in two dimensions have played an important role in number theory, differential and algebraic geometry, differential equations, and mathematical physics. Algebraic questions about surface group representations play a fundamental and unifying role. This workshop at AIM focused on various approaches to investigating surface group representations. Surface group representations are the points of certain "moduli spaces" which have a fascinating and rich geometry of their own.

The concept of moduli space is one of the great ideas of modern mathematics. Suppose you have a collection of objects. You think of each object as some point in an abstract space, and you form a geometric structure from that collection of points. The result is a moduli space, and that moduli space is of interest in its own right. For example, consider the set Š of all circles in the plane. Each circle is uniquely determined by (the coordinates of) its center and its radius. So to each circle we can uniquely associate a triple (x,y, r). In conclusion, the set

S = {(x, y, r): x ∈ R, y ∈ R, 0 < r ∈ R }

is a moduli space for the collection Š of all circles in the plane.

The workshop brought together researchers in numerous specialties to investigate the geometry on these moduli spaces from various points of view. A major general theme of the workshop was "how the moduli spaces look" and how their patterns of behavior changes as both the surface, and the geometry (as reflected in the Lie group) varies.