Five interpretations of Faà di Bruno's formula (Q2825651)

These lectures present the Faà di Bruno formula in five different settings; it also contains a short biography of Francesco Faà di Bruno and some historical remarks.NEWLINENEWLINEThis formula gives the \(n\)-th derivative of the composite function \(f\circ g\) in terms of the derivatives of the smooth functions \(f\) and \(g\).NEWLINENEWLINE{\parindent=0.6cm\begin{itemize}\item[--] In terms of groups: this formula appears in the composition of the group \(G^{\mathrm{dif}}\) of formal diffeomorphisms of the line. \item[--] In terms of Hopf algebras: it appears in the coproduct of the Hopf algebra \(H_{\mathrm{FdB}}\) of coordinates of the pro-algebraic group \(G^{\mathrm{dif}}\). This related it to other Hopf algebras, bases on rooted trees or Feynman graphs. Moreover, \(H_{\mathrm{FdB}}\) admits a (still mysterious) noncommutative version. \item[--] In terms of Lie algebras: Faà di Bruno formula is related to a Lie subalgebra of the Witt (or of the Virasoro) Lie algebra. By Cartier-Quillen-Milnor-Moore's theorem, this point of view is dual to the Hopf algebraic interpretation. This Lie algebra also admits extra structures, such as a prelie product or a brace bracket. \item[--] Combinatorially: \(H_{\mathrm{FdB}}\) is also the incidence algebra of a family of posets based on partitions. \item[--] In terms of operads: \(G^{\mathrm{dif}}\) is also a group associated to the operad Ass of associative algebras, and the Faà di Bruno formula reflects the composition of this operad. NEWLINENEWLINE\end{itemize}}NEWLINENEWLINEFor the entire collection see [Zbl 1318.16001].