Modular relations in pro-unipotent Grothendieck–Teichmüller theory

A conjectural description of the absolute Galois group of the rational field goes through the Grothendieck–Teichmüller group, introduced by Drinfeld in the 1990s. One can associate to both of these groups, as well as to several related objects, Lie algebras that are conjecturally isomorphic to each other. These algebras carry a natural filtration whose structure, still largely mysterious, is described by the Broadhurst–Kreimer conjecture: it predicts their graded dimensions in terms of the space of period polynomials of modular forms, which correspond to (cuspidal) cocycles of the modular group $\mathrm{PSL}_2(\mathbb{Z})$.

I will recall the definitions of these objects in an elementary way, and then state a theorem giving an explicit formula for the period polynomials of level-1 cusp forms. If time permits, I will also mention some ongoing work related to the Broadhurst–Kreimer conjecture in depth 4.