Symmetry: A really big idea

The concept of symmetry plays a strong role today in many of the exact sciences. Thus, for example, theoretical physicists, in searching for a unified field theory, have been led to the notion of supersymmetry, applied to the (super)strings, which, as some believe, are the fundamental building blocks of the Universe. Perhaps the foremost exponent of this position is the American physicist Edward Witten, of the Princeton Institute of Advanced Study, who, a few years ago, won a Fields Medal – the most prestigious award that can be given to a mathematician – for his fundamental theoretical contributions to superstring theory. Even more recently (August, 1998, at the International Congress of Mathematicians held in Berlin) the Cambridge mathematician Richard Borcherds was awarded a Fields Medal for his contribution to the development of symmetry theory, especially with respect to Witten theory and its relation to the sporadic finite groups.

What, then, is symmetry? In this chapter we attempt to make the idea precise, keeping our applications of the concept at a level where, as we hope, they will be appreciated by our readers. We will confine ourselves to the use of the symmetry concept within mathematics; and we must first of all emphasize that notions of symmetry, while fundamental to geometry, are certainly, and importantly, to be found in areas of mathematics outside geometry.