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.

My mathematical genealogy page has more info.

Important places for me have been, mathematically (besides my home base, Bogotá), first of all the Helsinki Logic Group (I have visited the University of Helsinki frequently, including for a whole year during my 2007 sabbatical; the blend of interests and the quality there are really at the top in the mixture of Model Theory, Set Theory and Philosophy), Chicago (University of Illinois at Chicago), Pittsburgh (Carnegie Mellon University, where I was a visiting professor for a year in 2002-2003), Jerusalem (my postdoc was at the Hebrew University of Jerusalem), Mexico City (UAM) and Cuernavaca (IMATE-UNAM). During the pandemic, I was also officially visiting the University of Torino (I taught Set Theory there during a sabbatical semester in 2021).

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).

Additionally, a strong interest in the philosophy of mathematics has evolved in participation in a series of events (Ongetemde Logica/Unfettered Logic – Aesthetics and Mathematics in Utrecht in 2007, Symposium on the Philosophy of the Logic of Sheaves in Cali, Colombia, in 2010, Simplicity, Ideals of Practice in Mathematics and the Arts at the CUNY Graduate Center in New York City in 2013, Mapping Traces: Categoricity, Representation, Definability in Bogotá in November of 2014; Getting There and Letting Go at the University of Helsinki in 2015; On The InfiniteAn Interdisciplinary Symposium at the Institut Henri-Poincaré in Paris in October of 2017.

With my colleagues Fernando Zalamea and John Alexander Cruz, we have organized Tertulias Matemáticas, a space of discussion on different aspects of mathematics, and on connections between Mathematics, Art and Philosophy.

Finally, interest in the (difficult) dialogue between Mathematics and Contemporary Art, we engaged in a project (moving topoi) together with mathematician Roman Kossak and two artists (Wanda Siedlecka and María Clara Cortés).

Click here for a detailed list of students who have written theses under my supervision, and the postdoctoral fellows I have hosted.