One of the main things that mathematicians do is to wrestle with problems of visualization. Typically the mathematician is trying to "visualize" a higher-dimensional object that does not live in our ordinary, 3-dimensional space. Yet there is a need to come to grips with this higher-dimensional, complex entity because it contains information that we must master and analyze.

One of the most powerful tools for the visualization process is symmetry. Symmetry allows one to reduce a complicated problem to a simpler one. It allows one to reduce the number of dimensions, and to actually "see" a version of what the object is. Representation theory, to be discussed below, is a way to study symmetry; it is one of the focuses of this workshop.

Symmetry is important in every branch of mathematics. Symmetry allows us to reduce a complex problem to simpler pieces. By analyzing those pieces, we can often reconstruct the original object.

The mathematical tool for capturing and describing symmetry is the group}. Invented by Evariste Galois and Augustin Cauchy in the early nineteenth century, groups now are used widely in mathematics, physics, and engineering. Axiomatically, a group is a collection G of objects together with a binary operation that is usually denoted by \ \cdot. We require that the group law be associative, that there be an identity element (called e), and that each element of the group have a multiplicative inverse. The collection of all integers, positive and negative, with the binary operation of addition, forms a group; in this group the identity element e = 0. The collection of 2 × 2 matrices having nonzero determinant, equipped with ordinary matrix multiplication, forms a group; in this group the identity element is the usual identity matrix with 1s on the diagonal and 0s elsewhere. It should be noted that, in practice in the subject of buildings and combinatorial representation theory, we think of a group from a functional point of view--it is the set of symmetries of some object.

One of the focuses of this particular workshop is group representations. A group representation hands you a new object for which the group is the set of symmetries. A class of groups that has been studied particularly effectively is those groups that can be perceived as a space tiled with mirrors that are arranged in a very symmetric manner. Simple group elements act as reflections in those mirrors.

Some tools that are used in representation theory come from geometry; among these are "buildings". Buildings, to be discussed in more detail below, incorporate our idea of space with many reflecting mirrors.

In classical geometry a standard problem was to construct a triangle with pre-specified side lengths. If the side lengths are a, b, c, then we understand that the conditions

a ≤ b + c ,

b ≤ a + c ,

c ≤ a + b

are both necessary and sufficient for the triangle to exist. See Figure 1. When the triangle lives in a more exotic geometric environment--say on a sphere--then in fact the conditions for a triangle with specified side lengths to exist (or even what it means to be a triangle) are much more complicated. Refer to Figure 2.

Figure 1

Figure 2

Mathematicians who study buildings and combinatorial representation theory try to smash together two given group representations to produce a third one. This process is strongly analogous to the classical problem of triangles (i.e., construcing the third leg of a triangle with specified side lengths), and produces similar diagrams and similar reasoning.

Another classical feature of the Euclidean geometry of triangles is scaling. If you stretch the side lengths of a triangle by a fixed amount then you get a larger (or smaller) triangle with the same shape. See Figure 3. In the analogy with representation theory, scaling does not necessarily hold. First of all, the only scalings that are valid in representation theory are of integer size. As a result, one cannot always scale up or down to a pre-specified scale. Buildings are a mathematical construct that encapsulate this triangular idea, and enable calculations of the sort indicated here. Buildings are spaces with reflecting mirrors replaced with multidimensional mirrors (that we saw in our discussion of representations). Just as a bathroom mirror can reveal hidden views, repeated views, and multiple views (sometimes even distorted views), the mirrors that we set up in buildings have not just two sides but (often) infinitely many sides. Buildings give us powerful geometric tools to study representation theory. For example, counting special triangles that live in buildings actually describes for us some very deep properties of representations.

Figure 3

Group theory lies at the foundation of much of algebra and analysis. It is part of the bedrock of modern mathematics. The language of symmetry, group theory serves many lines of inquiry as a powerful tool.