The philosophy of motives suggests the existence of a Galois theory for periods, which can be explicitly determined for certain families of periods such as hyperlogarithms. Recently, Abreu, Britto, Duhr, Gardi, and Matthew computed the Galois theory (a.k.a. the motivic coaction) of the coefficients in the epsilon-expansion of certain Feynman integrals in dimensional regularization, and observed that it could be packaged into succinct formulas at the level of power series. I will explain a proof of this phenomenon on the toy example of Lauricella hypergeometric functions, and suggest a geometric framework in which more general hypergeometric-type integrals are equipped with a Galois theory. This is based on the paper Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions written with Francis Brown, and ongoing work with Francis Brown, Javier Fresán, and Matija Tapušković.

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