Date : 5/2001

Référence : IHES/M/01/22

Résumé, Abstract :   ``Periods" is the generic term used to designate the numbers arising as integrals of algebraic functions over domains described by algebraic equations or inequalities with coefficients in $\Q$. This class of numbers, far larger and more mysterious than the ring of algebraic numbers, is nevertheless accessible in the sense that its elements are constructible and that one at least conjecturally has a way to verify the equality of any two numbers which have been expressed as periods. Most of the important constants of mathematics belong to the class of periods, and these numbers play a critical role in the theory of differential equations, in transcendence theory, and in many of the central conjectures of modern arithmetical algebraic geometry. The paper gives a survey of some of these connections, with an emphasis on explicit examples and on open questions.

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