Explore a comprehensive lecture on Weingarten calculus and its wide-ranging applications in mathematics and physics. Delve into the fundamental properties of compact groups and compact quantum groups, focusing on the existence and uniqueness of the Haar measure. Learn how Weingarten calculus systematically addresses the computation of moments in this context. Discover recent developments, theoretical properties of Weingarten functions, and their applications in random matrix theory, quantum probability, algebra, mathematical physics, and operator algebras. Examine topics such as polynomial functions on matrix groups, representation theoretic formulas, combinatorial formulations, and asymptotic behaviors. Investigate the connections to quantum information, matrix integrals, random tensors, and non-backtracking theory. Gain insights into this powerful mathematical tool through historical remarks, examples, and proofs presented by Benoit Collins in this 46-minute talk for the International Mathematical Union. Read more