Motivated by the importance of the Campbell, Baker, Hausdorff, Dynkin
Theorem 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: 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, provide a wide
spectrum of proofs from the modern literature, comparing different
techniques and furnishing a unifying point of view and notation, 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, give an outlook on the
applications, especially in Differential Geometry (Lie group theory) and
Analysis (PDEs of subelliptic type) and quickly enable the reader,
through a description of the state-of-art and open problems, to
understand the modern literature concerning 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 assumes some undergraduate-level knowledge of algebra and
analysis, but apart from that is self-contained. Part II of the
monograph is devoted to the proofs of the algebraic background. The
monograph may therefore provide a tool for beginners in Algebra.
