Watch a 58-minute mathematics lecture exploring the noncommutative maximal inequality and ergodic theorem for actions of amenable groups. Delve into the generalization of Birkhoff's pointwise theorem, examining both amenable group actions and noncommutative measure spaces. Learn how tools from Calderon-Zygmund theory are adapted to amenable groups, following a progression from basic concepts through advanced applications. Explore topics including maximal operators, noncommutative ergodic theory, martingale maximal inequalities, dyadic systems, and the geometry of local estimates. Understand the collaborative research between Léonard Cadilhac of Sorbonne Université and Simeng Wang of Harbin, which extends classical ergodic theory into more general mathematical frameworks.