mathematics research

Selected publications

Logic Textbook:

Lenguaje, teorías y modelos. Desde la incompletitud de Gödel hasta desarrollos contemporáneos. (With Jouko Väänänen.) Editorial EUDEBA. Buenos Aires, 2022.

«La lógica matemática hunde sus raíces más remotas en problemáticas provenientes del inicio mismo del indagar filosófico de la humanidad. Al mismo tiempo, florece como área plena de la matemática de manera relativamente reciente, en especial desde el inicio del siglo XX. De esta manera, la lógica es, a la vez, una disciplina intelectual antigua y un área muy contemporánea de la matemática. Una de sus subáreas importantes, la calculabilidad, ha resultado tener importancia fundamental en el desarrollo del aspecto más matemático de la computación. Otras partes de la lógica indagan matemáticamente sobre el rol final del lenguaje, su estudio como objeto matemático y su impronta en las estructuras del mundo. En esta obra, los autores explican las ideas necesarias de calculabilidad para el teorema de incompletitud de Gödel y luego proveen la demostración del teorema de completitud. Para ello siguen un orden inverso al usual: primero explican las razones filosóficas y matemáticas, luego dan una introducción a las ideas de la teoría de modelos moderna y, por último, ofrecen introducciones muy breves a temas importantes de la lógica matemática contemporánea como son la lógica de haces, la lógica infinitaria, la simbiosis, la lógica de segundo orden, las clases elementales abstractas y la lógica de la dependencia.»

Scientific Books (Monographs):

  • (Editor, with Juliette Kennedy and María Clara Cortés): Special volume for Theoria (Sweden), Mapping Traces. Vol 87 (2021). doi: 10.1111/theo12278. Editorial Introduction.
  • (Editor, with Roman Kossak, Juha Kontinen and Åsa Hirvonen): Logic Without Borders. A volume in honor of Jouko Väänänen’s Sixtieth Birthday. (Ontos Mathematical Logic, 5, De Gruyter.) In the last decades, mathematical logic has developed into a technically quite sophisticated area of mathematics. Nevertheless, inspirations from philosophy and computer science continue to be important and noticeable. The series publishes conference proceedings as well as monographs written by leading researchers in mathematical logic.
  • La tricotomía de Zilber: una breve introducción geométrica. (Zilber’s Trichotomy: a brief geometric introduction) Book in the series Publicaciones de la XXIV Escuela Venezolana de Matemáticas y EMALCA – Ediciones IVIC – Venezuela 2011. 95 pages.

Research papers:

  • The Small Index Property for Homogeneous Models in AEC’s. To appear in Archive for Mathematical Logic; a volume devoted to the proceedings of the IPM Conference on Set Theory and Model Theory, October 2015 (with Zaniar Ghadernezhad).
  • Uniqueness of Limit Models in Classes with Amalgamation. Math. Log. Quart. 62, No. 4–5, 367–382 (2016) (with Rami Grossberg and Monica VanDieren).
  • Sheaves of metric structures. Logic, Language, Information, and Computation (23rd International Workshop, WoLLIC 2016, Puebla, Mexico, August 16-19th, 2016. Proceedings, with Maicol Ochoa).
  • Around Stability Theory of Sheaves. 2014.
  • A global approach to AECs. (2014, with Hugo Luiz Mariano and Pedro Zambrano).
  • Around independence and domination in metric abstract elementary classes: assuming uniqueness of limit models. Math. Log. Quart. 60, No. 3, 211–227 (2014)  (with Pedro Zambrano).
  • Limit Models in Metric Abstract Elementary Classes: the Categorical case. (2013, with Pedro Zambrano).
  • The Two-Cardinal Problem, with Omitting Types (2013, with Luis Miguel Villegas).
  • Sheaves of G-structures and generic G-models. (2012, with Gabriel Padilla).
  • Hilbert Spaces with Generic Predicates. Accepted by Archiv for Mathematical Logic, undergoing modifications. (2011, with Alexander Berenstein).
  • Hrushovski fusions and Tame Abstract Elementary Classes. (2006, with Pedro Zambrano).
  • Notions of Independence for Metric Abstract Elementary Classes. The Royal Swedish Academy of Sciences – Mittag-Leffler Institute. Preprint series. Fall 2009. (With Pedro Zambrano)
  • Around Superstability in Metric Abstract Elementary Classes: Limit Models and r-Towers. The Royal Swedish Academy of Sciences – Mittag-Leffler Institute. Preprint series. Fall 2009. (With Pedro Zambrano).
  • Categoricity may fail late. Submitted to Journal of Symbolic Logic. 2008. (With Saharon Shelah).
  • Laver Diamonds on Strongly Unfoldable Cardinals. (2003, with Franqui Cárdenas).
  • Beautiful pairs in continuous model theory. (In preparation, with Alexander Berenstein).
  • On categoricity spectra for classes with no maximal models II. (In preparation, with Saharon Shelah).
  • Toward categoricity for classes with no maximal models. Annals of Pure and Applied Logic 97, 1999, 1–25 (with Saharon Shelah).
  • Heights of Models of ZFC and the Existence of End Elementary Extensions II. The Journal of Symbolic Logic 64-3, Sept. 1999.
  • Chains of End Elementary Extensions of Models of Set Theory. The Journal of Symbolic Logic 63-3, Sept. 1998.

Reviews:

Surveys:

  • La tensión entre teoría de modelos y análisis matemático: estabilidad y la exponencial compleja. Boletín de Matemáticas; Vol. 11, núm. 2 (2004); 95-108
  • \aleph_\omega de aleph a omega. Boletín de Matemáticas. IV – 1. June 1997.
  • La Conjetura de Łoś: un teorema aún central en Teoría de la Clasificación. Boletín de Matemáticas. IV – 1. June 1997.
  • Heights of Models of ZFC and the Existence of End Elementary Extensions, in Models, Algebras and Proofs, eds: Xavier Caicedo and Carlos Montenegro (Tenth Latin American Symposium in Mathematical Logic, Bogotá, 1995), 1999.

Other:

  • Memorial Tribute: In Memory of Ken Kunen (with Mirna Džamonja, Joan Hart and Andrea Medini). November 2022. With contributions by Alan Dow, Michael Hrušák, Stephen Jackson, István Juhász, H. Jerome Keisler, Steffen Lempp, Donald Martin, Adrian Mathias, Jan van Mill, Arnold W. Miller, Justin Moore, Dilip Raghavan, John Steel, Frank Tall, and Hugh Woodin.

Course and Seminar Notes:

Lectures: click here for lectures between 2012 and now.

Lectures from 2012 and older:

  • II Encontro de Topologia e Teoria dos Conjuntos, Salvador de Bahia, February 2012.
  • Lecture “Some Set-Theoretic Dichotomies in Model Theory”, Kobe, Japan, January 2012.
  • Lecture for Logic Seminar “Dicotomies en théorie des modèles”, Lyon, Francia, November 2011.
  • Infinity Conference – Centre de Recerca Matemàtica – Bellaterra, Catalonia, July 2011.
  • Stability Theory of Polish Spaces, Helsinki, June 2011.
  • Philosophy Symposium on the Logic of Sheaves, within the Colombian Congress of Philosophy – Cali, Colombia, October 2010.
  • The Model Theory of Sheaves (minicourse) – Jouko Väänänen’s Sixtieth Birthday Conference. Helsinki, September 2010.
  • Barcelona Logic Seminar. September 2010: On type decomposition in NIP theories.
  • Infinity Conference – Centre de Recerca Matemàtica – Bellaterra, Catalonia, 2010: Around forcing categoricity of a bipartite random graph, at \aleph_2.
  • Mittag-Leffler Institute – Octubre de 2009: \kappa-dependent AECs.
  • European Science Foundation – Model Theory – Mathematical Research and Conference Center – Będlewo – Polonia. July 2009.
  • Kunen Fest: Topology and Set Theory Conference, april 2009, Madison, Wisconsin: Metric Continuous Abstract Elementary Classes Why? Superstability, NIP, examples.
  • Logic Seminar, Universidad Autónoma Metropolitana – Mexico City – October 2008 – Clases Elementales Abstractas e invariantes geométricos.
  • Logic Seminar, University of East Anglia, Norwich, Inglaterra, July 2008. Continuous Model Theory – Some simplicity issues.
  • XIV SLALM – Paraty, Brasil: Continuous Model Theory, Metric AECs: Simplicity issues, expansions by predicates. May 2008.
  • Logic Seminar, Università di Torino: Abstract Elementary Classes: A middle ground of generalization. December 2007.
  • Logic Seminar, Universidad de Barcelona, november 2007: Expansions amb predicats en lògica continua: Estabilitat i simplicitat.
  • Aesthetics and Mathematics Symposium, Universidad de Utrecht (The Netherlands), november 2007: Higher Amalgams: the wonderful mélange.
  • Southern Wisconsin Logic Colloquium, Madison, Wisconsin, october 2007: Generic Predicates in Continuous Model Theory.
  • Lecture for the American Mathematical Society Sectional Meeting, Chicago, October 2007: Around simplicity in various non-elementary classes.
  • Colombian Congress of Mathematics, 2007: Teoría de Modelos de Espacios de Hilbert: pares buenos y menos buenos. Cleas versus gatos.
  • Logic Seminar, Université Claude-Bernard Lyon-I (France), june 2007: Espaces de Hilbert et Prédicats Génériques: deux exemples et quelques questions.
  • Helsinki Logic Seminar, june 2007: On Simplicity in Continuous Model Theory.
  • (towards a) Model Theory for Chemical Reaction. Logic Seminar. University of Helsinki, April 2007.
  • Generic Predicates over Hilbert Spaces – (Continuous) Model Theory of Pairs. Colloquium on Mathematical Logic, Universiteit van Amsterdam. March 2007.
  • Abstract Elementary Classes and Finite Combinatorics. Finite Model Theory Seminar, University of Tampere, Finland, February 2007.
  • Generic Predicates in Continuous Model Theory. Logic Seminar. University of Helsinki, February 2007.
  • Classifying mathematical structures; some trends in model theory, colloquium talk at Carnegie-Mellon University, Pittsburgh, April 2002.
  • On abstract elementary classes, invited talk at the logic seminar, University of Illinois at Chicago, November 2001.
  • The weak expressive power of First Order Logic in Mathematics, and the strength of this limitation, invited talk at the joint mathematics and philosophy symposium, Carnegie-Mellon University, Pittsburgh, October 2001.
  • Categoricity theorems may fail for abstract elementary classes, invited talk at the MAMLS Meeting, Rutgers, February 2001.
  • Lógica en Álgebra y Análisis: caminos para sacarle jugo a las limitaciones expresivas, invited colloquium talk at Universidad de los Andes, October 2000 (in Spanish).
  • Categoricity theorems may fail for abstract elementary classes, contributed talk at the European Logic Colloquium, Paris, 2000.