Explore a groundbreaking generalization of the persistence algorithm for decomposing multi-parameter persistence modules in this 54-minute lecture from the Applied Algebraic Topology Network. Delve into the challenges of extending the classical persistence algorithm to multi-parameter settings and discover a novel approach based on generalized matrix reduction techniques. Learn about the improved time complexity of this new algorithm compared to existing methods like the Meataxe algorithm. Examine the connections between graded modules from commutative algebra and matrix reductions, and gain insights into persistent graded Betti numbers and block code. The lecture covers introductory concepts, examples, computation of decompositions, diagonalization, linearization, and concludes with a discussion on future work in this emerging area of topological data analysis.