Periods and the conjectures of Grothendieck and Kontsevich-Zagier (Q461503)

The first part of this paper is an introduction to the periods, the conjecture of \textit{M. Kontsevich} and \textit{D. Zagier} [in: Mathematics unlimited -- 2001 and beyond. Berlin: Springer. 771--808 (2001; Zbl 1039.11002)], the conjecture of Grothendieck on the relation between the transcendence degree of fields of periods and the dimension of the motivic Galois group, the extension of Grothendieck's conjecture by \textit{Y. André} to base field of nonzero transcendence degree [Une introduction aux motifs. Motifs purs, motifs mixtes, périodes. Paris: Société Mathématique de France (2004; Zbl 1060.14001)], and the connection, due to Kontsevich, between these conjectures [\textit{A. Huber} and \textit{S. Müller-Stach}, ``On the relation between Nori motives and Kontsevich periods'', preprint, \url{arXiv:1105.0865}].NEWLINENEWLINEThe second part of this paper is devoted to the geometric version of the conjecture of Kontsevich-Zagier, of which the author has provided a proof [\textit{J. Ayoub}, Ann. Math. 181, No. 3, 905--992 (2015; \url{doi:10.4007/annals.2015.181.3.2})].