Motivated by the importance of the result usually referred to as the Theorem of Campbell, Baker, Hausdorff and Dynkin in many different branches of Mathematics and Physics (Lie group-Lie algebra theory, linear PDEs, Quantum and Statistical Mechanics, Numerical Analysis, Theoretical Physics, Control Theory, sub-Riemannian Geometry), this monograph is intended to:§1) fully enable readers (graduates or specialists, mathematicians, physicists or applied scientists, acquainted with algebra or not) to understand and apply the statements and numerous corollaries of the main result;§2) provide a wide spectrum of proofs from modern literature, comparing the different techniques and furnishing a unifying point of view and notation;§3) provide a thorough historical background of the results, together with unknown facts about the effective early contributions by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff and Dynkin.§4) predict applications, especially in Differential Geometry (Lie group theory) and Analysis (PDEs of subelliptic type);§5) quickly enable the reader, through a description of the state-of-art and open problems, to understand the modern literature on a theorem which, though having its roots in the beginning of the 20th century, has not ceased to provide new problems and applications.§The book is self-contained (apart from the use of some undergraduate algebra and analysis). Part II of the monograph is devoted to the proofs of the algebraic background: It is left to the reader to decide which proofs to consult or to skip. The monograph may therefore provide a valuable tool for beginners in algebra.
