This talk was originally inspired by Zagier's construction of single-valued polylogarithms and regulators in the late 80's.

In short, single-valued functions are ubiquitous in mathematics and physics, since well-defined problems have well-defined answers. On the other hand, the solution to such a problem is often given by an integral, which is usually a multi-valued function of its parameters. The reason is that integration is a pairing between differential forms and chains of integration, and the latter are ambiguously defined.

In this talk, which is joint work with Clément Dupont, I will describe a way to pair differential forms with `duals of differential forms'. This defines a notion of integration which satisfies the usual rules, but is automatically single-valued. Many well-known constructions in mathematics and physics are examples of such objects. Depending on time I will mention some of the following applications: non-archimedean height pairings, single-valued versions of multiple zeta values, new classes of non-holomorphic modular forms, string theory amplitudes and double-copy constructions; all of which intersect with the work of Zagier in some way.

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