The first part will be historical, recounting how Misha Zhitomirskii and I came upon what we called the Monster Tower through a question in singularity theory. Later Alex Castro uncovered that Colley, Kennedy, and their crew had already been climbing around for a while in this same tower. They called it the Semple Tower and had a different aim in mind and consequently looked at it from the perspective of a different (smaller) group acting on the tower than our group, the group of contact diffeomorphisms of a contact 3-manifold. The primary discrete invariant of these group actions is the RVT code (shifted down by one for C--K) which is almost the same thing as the Puiseux characteristic of a class of plane curves which is attached to points of the tower. (The data encoded by the Puiseux characteristic is the same as that encoded by the equisingularity class or numerical semigroup of a curve.)

The second part of the talk will describe how screams issuing from the tower alerted us to the likely existence of “complete secondary invariants.” In the case of C--K these secondary invariants are the value sets as first described by Zariski and the theorem that they are complete is due to Hefez and Hernandes. The clues --- the noticing of the screams --- were brought into the open by the recent thesis of Justin Lake, co-advised with Gary and Lee McEwan.