In complex function theory, a holomorphic (or analytic) function on an open set U ⊆ C is a function f: U → C which has a convergent power series expansion about each point p ∈ U:

f(z) = Σ_j=0^∞ a_j (z - p)^j .

Alternative characterizations of holomorphic functions may be formulated in terms of the Cauchy-Riemann equations, or in terms of the geometric property of conformality.

It was a brilliant idea of Bernhard Riemann (1826--1866) to conceive of a geometric surface that would capture the analytic properties of such a function f. In his honor, these surfaces are now called Riemann surfaces. Figure 1, for example, shows the surface for the function log z. The spiral reflects the fact that log (x + iy) is only defined up to an additive factor of 2kπ i, k ∈ Z. The function "square root z" has a 2-sheeted Riemann surface, while the function "mth root z" has an m-sheeted Riemann surface. Every holomorphic function has its own characteristic Riemann surface and vice versa. Some---indeed the most interesting---Riemann surfaces have holes and punctures and singularities in them. These are the focus of the present workshop.

Figure 1

Let R be a Riemann surface with holes and punctures. The goal is to understand, in an analytic manner, the geometry of such a surface. We begin by considering consider the first homotopy group of R. This group consists of equivalence classes of continuous maps φ : [0,1] → R, where we require that φ(0) = φ(1) = q for some fixed point q ∈ R;, thus the image of the map is a closed loop passing through q. Two such maps are "equivalent" if they can be continuously deformed to each other. The collection of equivalence classes of such loops can be made into a group with the binary operation of juxtaposition. Namely, if φ₁ and φ₂ are two such mappings (or loops) then we form the new loop φ₂ × φ₁ by following φ₁ by φ₂ We call this group the first homotopy group of R and we denote this group by π₁(R).

If N is a positive integer then we let GL(N) denote the group of N × N invertible matrices (i.e., matrices with nonvanishing determinant). The key idea now is to consider homomorphisms Φ: π₁(R) → GL(N). We call such a function a representation of the homotopy group. We understand the structure of R by understanding certain of these maps Φ. We construct especially useful maps Φ by isolating particular elements of GL(N) and then forcing the generators of π₁(R) to be mapped to those elements. One of the principal tools in this subject is to consider the "moduli space" M of all such mappings. Here a moduli space is an abstract idea of a geometric surface or construct that parametrizes all such maps. [Another important idea in Riemann surface theory is the idea of Teichmüller space. A Teichmüller space is a moduli space for all compact Riemann surfaces.] This moduli space is in fact the "character variety" referred to in the title of this workshop.

The moduli space M is a geometric object that now can be studied in its own right. We may apply combinatorial techniques and ask questions of the following type:

• How many connected components does the moduli space have?
• What singularities does the moduli space have?
• What is the cohomology (i.e., what types of holes, and of what dimensions) of the moduli space?

The cohomology groups of the moduli space have associated Poincaré polynomials. It is conjectured---and this is one of the big open questions in the subject---that these polynomials are orthogonal in a natural inner product.

It turns out that the polynomials described in the last paragraph are Macdonald polynomials. Macdonald polynomials arise very naturally in a variety of contexts in algebra and representation theory. The present mathematical context gives a conjectured geometric interpretation of these polynomials, and that is a matter of great interest.

Like many AIM workshops, this one brings together workers from a variety of disciplines, bringing with them many different points of view. A number of these mathematicians wish to learn about Macdonald polynomials, and how they can be applied to understand geometric situations. The result has been a useful cross-fertilization of different fields, and the creation of a good deal of new mathematics.