Monster towers from differential and algebraic viewpoints

Monster Towers were fathered by J. G. Semple in 1954 and were frequently used in algebraic geometry since then. Those towers got a second youth with A. Kumpera's works in the 1980's and Bryant and Hsu's modern treatment (1993) of the Cartan prolongation in differential geometry. In consequence, there have emerged Goursat- and [special-multi]-flags living on the stages of the same Monster/Semple Towers, and featuring rich trees of singularities (albeit not of a wild functional type, omnipresent among more generic structures in the tangent bundles to manifolds). Now a unification of the two approaches is in sight. In fact, after the works of Castro et al., (2017), Mormul and Pelletier (2020), and recently of the present author (2020), there emerges a clear two-way dictionary allowing one to quickly interpret algebraically defined singularities in Semple Towers in differential terms, and also to make readable to algebraic geometers differential constructions done long since in Monster Towers.