Kontsevich–Zagier's period conjecture

A period is a complex number represented by an integral of an algebraic differential form over a domain defined by algebraic equations and inequalities, with algebraic coefficients. Suppose a period has two different such integral representations. Kontsevich–Zagier's period conjecture. One expression can be transformed into the other by applying the three integral transformation rules of additivity, change of variables, and Stokes' formula, with all integrands and domains of integration algebraic with algebraic coefficients.

The conjecture concerns the equality of explicit period representations and would provide a complete set of algebraic rules for proving such equalities. The source presents the problem of explicitly exhibiting a number outside the ring of periods as another fundamental open problem.