Hopf algebra structures in the cohomology of moduli spaces

I will discuss two spectral sequences of Hopf algebras, a morphism between them, and some applications. One spectral sequence is related to the algebraic K-theory of the integers and was introduced, without Hopf algebra structure, by Quillen in his proof that algebraic K-groups are finitely generated. The other spectral sequence is related to the Grothendieck-Teichmüller group and to Kontsevich's graph complexes. This is joint work with Francis Brown, Melody Chan, and Sam Payne, cf. arXiv:2405.11528.