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).
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 -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 Infinite – An Interdisciplinary Symposium at the Institut Henri-Poincaré in Paris in October of 2017.
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
Former postdocs I hosted:
- Zaniar Ghadernezhad (2013-2014).
- Sonat Süer (2008-2009).
Current and former students – and some of their theses.
Doctoral students.
- José Nicolás Nájar – since 2020.
- Óscar Pérez (Philosophy doctoral student) – since 2020.
- Edwin Rodrigo Celis Montealegre – since 2020.
- Camilo Enrique Argoty Pulido (coadvisor A. Berenstein), Ph. D. in Mathematics, 2015. Model Theory of representations of operator algebras.
- Pedro Hernán Zambrano Ramírez, Ph. D. in Mathematics, 2012. Around Superstability in Metric Abstract Elementary Classes.
Master’s students.
- Óscar Javier Pérez Lora, M. A. in Philosophy, 2020. La teoría de modelos como expresión de una Teoría general de los enlaces entre esencia y existencia en la filosofía matemática de Albert Lautman. Mención meritoria.
- Juan Ignacio Agudelo Ramírez (current, since 2018).
- David Valderrama Hernández (current, since 2019).
- Nicolás Medina Sánchez, M. Sc. in Applied Mathematics, 2019. Los Haces como Espacios de Estado. Mención meritoria.
- Johan Felipe García Vargas, M.Sc. in Mathematics, 2017. Internalidad: Un camino de la Teorı́a de Modelos hacia la Teorı́a de Galois.
- Felipe Posada, 2014-2016.
- Maicol Arley Ochoa Daza, M. Sc. in Mathematics, 2012. Sheaves of Topological Structures over the Partial Order of Finite Dimensional Subspaces of a Hilbert Space
- María Victoria Cifuentes Amado. M. Sc. in Mathematics, 2012. Clases de Fraïssé, Construcciones de Hrushovski y Algunas Aplicaciones de la Teoría de Ramsey. Mención meritoria.
- Rafael Eduardo Benjumea Hoyos (coadvisor: Carlos Di Prisco (IVIC), degree from Universidad Nacional de Colombia). 2007. Preservación de modelos de Solovay bajo el forcing de Sacks.
- Pedro Hernán Zambrano Ramírez, M. Sc. in Mathematics, 2005. Construcciones de Hrushovski y clases no elementales.
- Paul Baginski (Master’s Honors Program, Carnegie Mellon University). 2003. The Jet-Space Proof of the Mordell-Lang Conjecture in Characteristic Zero.
- Franqui Solís Cárdenas Poloche, M. Sc. in Mathematics, 2002. Lanzamiento de un club a un cardinal desdoblable.
Undergraduate students.
(In Bogotá, undergraduate students may write a monograph – trabajo de grado – to start their path toward more mathematics.)
- Nicolás Martínez Bejarano, Philosophy, 2018. Imagen que cambia, imagen que permanece. Estudio sobre el papel de la imagen en el tránsito del ciclo cuaternario al ciclo ternario de Ramón Llull. Premio Mejor Tesis de Pregrado, 2018.
- David Valderrama Hernández, Mathematics, 2018. Independencia y Calculabilidad.
- Edwin Rodrigo Celis Montealegre, Mathematics, 2018. Categoricidad de los espacios de Banach.
- Sebastián Ricardo Cristancho Sierra. Mathematics, 2018. Satisfacción y poder expresivo: algunos resultados sobre sistemas lógicos abstractos.
- Juan David Cárdenas Ramírez, Mathematics, 2018. Aspectos matemáticos de la danza contemporánea.
- Juan Camilo Arosemena Serrato. Mathematics, 2014. La propiedad del árbol en cardinales sucesores.
- Miguel Cardona, Mathematics (degree from Universidad Nacional, sede Medellín). 2013. En torno a Categoricidad y Estabilidad en Teoría de Modelos.
- Alejandra Rincón Hidalgo, Mathematics, 2010. Clases elementales abstractas inducidas por clases Cotilting.
- Fabián Agudelo. Mathematics, 2011. Conjuntos externamente definibles y pares dependientes.
- Joaquín Maya Duque, Mathematics, 2010. Una axiomatización del Cubrimiento universal de una variedad proyectiva compleja.
- Diana Carolina Montoya Amaya, Mathematics, 2008. Versiones Definibles en Teorı́a Combinatoria de Conjuntos.
- Miguel Jara, Mathematics, 2006. Algunos teoremas de no-estructura para teorı́as no clasificables.
- Rafael Eduardo Benjumea Hoyos. Mathematics, 2005. Pcf y aplicaciones.
- Juan Diego Caycedo Casallas, Mathematics, 2005. Interpretación de Grupos en Teorı́a de Modelos Homogénea.
- Mario Andrés Velásquez Méndez, Mathematics, 2005. Aplicaciones de los pares de Vaught en contextos modelo-teóricos.
- Hermes Jackson Martínez Navas, Mathematics, 2003. Algunas conexiones entre Estabilidad en Clases No Elementales y estudio de Estructuras Analíticas.
- Javier Arturo Moreno Camargo, Mathematics, 2001. Teoría de Modelos Geométrica y Aplicaciones. Mención meritoria. Premio mejor tesis de pregrado 2001.
- Francisco Vargas Mancera, Mathematics, 2000. Mención meritoria.