III.60 Moduli Spaces

An important general problem in mathematics is classification (see THE GENERAL GOALS OF MATHEMATICAL RESEARCH [I.4 §2]). Often, one has a set of mathematical structures and a notion of equivalence, and one would like to describe the EQUIVALENCE CLASSES [I.2 §2.3]. For example, two (compact, orientable) surfaces are often regarded as equivalent if each can be continuously deformed into the other. Each equivalence class is then fully described by the GENUS [III.33], or “number of holes,” in the surface.

Topological equivalence is rather “crude,” in the sense that it is relatively easy for two surfaces to be equivalent. As a result, the equivalence classes are parametrized by a fairly simple set: the set of all positive integers. But there are many geometrical contexts in which finer notions of equivalence are important. For example, in several contexts one wishes to regard two two-dimensional LATTICES [III.59] as equivalent if one is a rotation and enlargement of the other. Equivalence relations such as this one often lead to parameter sets that themselves have an interesting geometrical structure. Such sets are called moduli spaces. For details, see [IV.8] and also [V.23].