I Teoremi di Campbell, Baker, Hausdorff e Dynkin. Storia, Prove, Problemi Aperti.

The aim of this lecture is to provide an overview of facts and references about past and recent results on the Theorem of Campbell, Baker, Hausdorff and Dynkin (shortcut as the CBHD Theorem), following the recent preprint monograph [13]. In particular, we shall give sketches of the following facts: A historical précis of the early proofs (see also [1]); the statement of the CBHD Theorem as usually given in Algebra and that employed in the Analysis of linear PDE’s; a review of proofs of the CBHD Theorem (as given by: Bourbaki; Hausdorff; Dynkin; Varadarajan) together with a unifying demonstrational approach; an application to the Third Theorem of Lie (in local form). Some new results will be also commented: The intertwinement of the CBHD Theorem with the Theorem of Poincaré-Birkhoff-Witt and with the free Lie algebras (see [12]); recent results on optimal domains of convergence.