## 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.

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).

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 in 2010, Simplicity, Ideals of Practice in Mathematics and the Arts at the CUNY Graduate Center in New York City in 2013,  and now organizing Categoricity, Representation, Definability in Bogotá in November of 2014.

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).

### Hosting of postdoctoral fellows

I have hosted some postdocs in Bogotá:

• Sonat Süer (2008-2009)

### Current and former students

#### Doctoral students.

• Camilo Argoty (coadvisor Alexander Berenstein) – current.
• Pedro Zambrano (Ph. D. in Mathematics, 2012).

#### Master’s students.

• Johan García – current.
• Maicol Ochoa. 2012.
• María Victoria Cifuentes. 2013.
• Rafael Benjumea (coadvisor Carlos Di Prisco, degree from Universidad Nacional de Colombia). 2008.
• Paul Baginski (Master’s Honors Program, Carnegie Mellon University). 2003.
• Franqui Cárdenas. 2001.

(In Bogotá, undergraduate students may write a monograph – trabajo de grado – to start their path toward more mathematics.)

• Camilo Arosemena (coadvisor Carlos Di Prisco). 2014.
• Miguel Cardona (degree from Universidad Nacional, sede Medellín). 2013.
• Joaquín Maya. 2012.
• Alejandra Rincón. 2011.
• Fabián Agudelo. 2011.
• Diana Montoya. 2008.
• Miguel Jara. 2006.
• Rafael Benjumea. 2005.
• Juan Diego Caycedo. 2004.
• Hermes Martínez. 2004.
• Javier Moreno. 2001.
• Francisco Vargas. 2000.