## between model theory and set theory (via geometry, philosophy and now art)

I started this path now a long time ago – I worked with Xavier Caicedo in Bogotá on the Model Theory on Sheaves first, then switched to Set Theory during my Ph. D. with Ken Kunen in Madison (really, in retrospect, it was about Model Theory of models of Set Theory – but the sort of “invariants” that came about were very set-theoretic in nature (large cardinals – unfoldable cardinals, strong and long unfoldables, and all sort of combinatorial properties connecting them to the daunting task of classifying models of set theory!)).

The path continued: next stage was Jerusalem: postdoctoral work with Saharon Shelah. Although the initial work was connected to Borel Sets with Large Squares, I promptly jumped back to Model Theory (my first mathematical love) and started the long way toward the Model Theory of Abstract Elementary Classes. This enormous task (lifting a lot of Model Theory – Stability and Classification Theory – to the much wider realm of AECs) has a fascinating blend of between Model Theory and Set Theory, doing a lot of Model Theory when only weak remnants of compactness (such as amalgamation properties) are present, and extending Categoricity, Stability, NIP, etc. – revealing deep model-theoretic facts that link the behavior of first order theories with many other features.

Currently I am engaged in continuing the classification theory of AECs, combining it with work in the Model Theory of Sheaves, and applications of both to Number Theory (the Model Theory of $j$-invariants), non-commutative geometry (the Model Theory of the field of characteristic one).