The Betti side of the double shuffle theory: a survey

This is a survey of arXiv:1803.10151v4, arXiv:1807.07786v2 and arXiv:1908.00444v2 by H. Furusho and the author. The purpose of this series of papers is: (1) to give a proof that associator relations imply double shuffle relations, alternative to Furusho's paper arXiv:0808.0319v3; (2) to make explicit the bitorsor structure on Racinet's torsor of double shuffle relations. The main tool is the interpretation of the harmonic coproduct in terms of the topology of the moduli spaces $\mathfrak M_{0,4}$ and $\mathfrak M_{0,5}$, introduced in Deligne and Terasoma's 2005 preprint, and its extension to the Betti setup.