Noncommutative Topology: Spaces

Abstract

The geometrical study of quadratic curves or surfaces, i.e., zero sets of second-degree polynomials, proceeds by examining points of intersection or tangent lines directly; but already for cubic curves it pays to examine first the ideal of all polynomials that vanish on the curve: in this way the study of an algebraic variety (the zero set of a given finite collection of polynomials) is replaced by the study of the corresponding polynomial ideal. Such a fundamental geometrical object as an elliptic curve is best studied not as a set of points (a torus) but rather by examining functions on this set, specifically the doubly periodic meromorphic functions: Weierstrass opened up a new approach to geometry by studying directly the collection of complex functions that satisfy an algebraic addition theorem, and derived the point set as a consequence 51.