Explore the concept of lower sets in products of well-ordered sets in this 33-minute lecture by Harry Altman. Delve into the question posed by Aschenbrenner and Pong regarding the type (maximum linearization) of the set of lower sets of Nm. Examine the generalization to lower sets and bounded lower sets in products of larger ordinals, and consider a related question about weakly decreasing sequences in well partial orders. Learn about well partial orders and their maximum linearizations, finite multisets, and the proof sketch for DIN. Understand the general formula for the core case, solve the recursion, and explore the surreal exponential. Conclude with a conjecture related to the two-dimensional case in this advanced mathematical exploration presented at the Hausdorff Center for Mathematics as part of the Hausdorff Trimester Program: Types, Sets and Constructions.