Weingarten Calculus and Its Applications

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.

Intro
Contents
The Haar measure on compact groups
Polynomial functions on a matrix group
Fundamental integration formula
Historical remarks and comments
Representation theoretic formulas (unitary case)
Combinatorial formulations
Digression: the quantum group case
Leading order Asymptotics of Wg (U, case)
Applications of the asymptotics (a subjective selection)
Asymptotic freeness (pointwise, leading order)
Asymptotic freeness: quantum (pointwise, leading order)
Quantum Information (pointwise, leading order)
Higher order asymptotic freeness (higher order)
Matrix integrals and random tensors (higher order)
Uniform estimates
Centered version
Strong Asymptotic freeness Centering
Outline of the proof
Non-Backtracking theory
Concluding remarks