Explore the concept of stability manifolds and their role in homological structures of triangulated categories in this lecture by Daniel Halpern-Leistner from Cornell University. Delve into the proposed "noncommutative minimal model program" and its implications for defining canonical decompositions of derived categories. Examine the partial compactification of stability manifolds through augmented stability conditions and the introduction of multi-scale decompositions. Investigate the relationship between multi-scale lines and multi-scale differentials in dynamics. Consider the main conjecture regarding the space of augmented stability conditions as a manifold with corners and its potential consequences for moduli spaces of semistable objects in smooth and proper dg-categories.