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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.8 Moduli Spaces

David D. Ben-Zvi

Many of the most important problems in mathematics concern CLASSIFICATION [I.4 §2]. One has a class of mathematical objects and a notion of when two objects should count as equivalent. It may well be that two equivalent objects look superficially very different, so one wishes to describe them in such a way that equivalent objects have the same description and inequivalent objects have different descriptions.

Moduli spaces can be thought of as geometric solutions to geometric classification problems. In this article we shall illustrate some of the key features of moduli spaces, with an emphasis on the moduli spaces of RIEMANN SURFACES [III.79]. In broad terms, a moduli problem consists of three ingredients. ...

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