Shifted Substitution in Non-Commutative Multivariate Power Series with Applications to Free Probability Theory

Explore a mathematical lecture examining group laws on formal power series in non-commuting variables through their interpretation as linear forms on graded connected word Hopf algebra. Delivered at the Workshop on "Recent Perspectives on Non-crossing Partitions through Algebra, Combinatorics, and Probability" at the Erwin Schrödinger International Institute, discover how left-linear group laws associate with pre-Lie structures on formal power series. Learn how these mathematical structures provide a group theoretic framework for understanding identities and transformations central to non-commutative probability theory, particularly in Voiculescu's free probability theory. Gain insights into shifted substitution concepts and their applications in multivariate power series during this 30-minute advanced mathematics presentation.