Exploring Moduli Spaces

Abstract

There's more to math than just numbers and equations. In fact, much of mathematics concerns space?not outer space, but mental representations of space and spatial relationships. While many people think of Descartes as the first person to describe points in a plane by means of a pair of coordi nates, the idea of coordinatizing planar space was known to Hipparchus in 150 B.C. And by 1640, Pierre de Fermat had already developed the idea of expressing a curve in the plane as the solution set of an equation in x and);. Rene Descartes, however, was the first to systemically use the coordinate plane to turn problems of geometry into problems of algebra, and vice versa. Today, at the very center of modern research in number theory and geometry lies a much-generalized form of Descartes's idea?that of moduli spaces.