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MATHEMATICS

Para luchar contra el pragmatismo y la horrible tendencia a la consecución de fines útiles, mi primo el mayor propugna el procedimiento de sacarse un buen pelo de la cabeza, hacerle un nudo en el medio y dejarlo caer suavemente por el agujero del lavabo.

"Pérdida y recuperación del pelo", Julio Cortazar

I am a model theorist. An excellent undergraduate course in logic taught by Andrés Villaveces in the Universidad Nacional de Colombia, introduced me to the metamathematical universe of abstract structures and formal languages and since then I've been most of the time immersed on it. Currently, I am math graduate student in UIUC finishing his PhD and looking for a job. My advisor is Anand Pillay. My specific interests include geometric model theory and the connections between model theory and differential algebra (hopefully with some geometrical flavor). At the moment, I'm devoted to the study of Differential Galois Theory from a model theoretical perspective.

Current Work

My thesis (04/21/08 Final Version), finally finished on Fall 2007, introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat and van der Put. Instead of taking an algebraic approach, we use the methods and framework provided by the model theory of iterative differential fields. After defining what we mean by a strongly normal extension of iterative differential fields, we prove that these extensions have good Galois theory and that a primitive element theorem holds. Then, making use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations.

Future Projects

As a follow up of what I did in my thesis, I'd like to explore the possibility of developing a generalized strongly normal theory for positive characteristic ID-Fields. This would allow us to do Galois theory of iterative logarithmic differential equations on algebraic groups not necessarily over the constants. I'm also interested in the study of difference-differential equations (and their Galois theory) over fields of positive characteristic.

I'm applying to some postdocs to continue working on these issues.

UIUC Courses & Seminars

These are the courses I have taken or I'm planning to take here in UIUC:

F2001
  • Math 402 (A Second course in Abstract Algebra II) with Professor Philiph Griffith.
  • Math 414 (Advanced Topics on Logic: Model Theory in Functional Analysis) with Professor C. Ward Henson.
S2002
  • Math 405 (Algebraic Number Theory) with Professor Marcin Mazur.
  • Math 422 (Algebraic Geometry) with Professor Iwan Duursma.
  • Math 490 (Reading Course: Stability Theory) with Professor Anand Pillay.
F2002
  • Math 423 (Differentiable Manifolds) with Professor Eugene Lerman.
  • Math 441 (Real Analysis) with Professor Marius Junge .
  • Math 490 (Reading Course: Algebraic Geometry and Model Theory) with Professor Anand Pillay.
  • Math 414 (Advanced topics in Logic: Simple Theories ) with Yevgeniy Vasilyev.
S2003
  • Math 412 (Computability Theory) with Professor Carl Jockusch.
  • Math 440 (Complex Analysis) with Professor Joseph Miles.
  • Math 490 (Reading Course: More applied Model Theory) with Professor Anand Pillay.
  • Math 414 (Advanced Topics in Logic: Stability Theory) with Professor Anand Pillay.
Sum
2003
  • Comps (Holy crap! I'm done with comps!)
  • Seminar (Interpretability of Groups) Organized by me with the advice of Professor Anand Pillay.
F2003
  • Math 416 (Adv. Topics in Abs. Algebra: Riemann Surfaces and Algebraic Curves) with Professor Dror Varolin.
  • Math 499 (Thesis Research: Prelim preparation) with Professor Anand Pillay.
  • Math 414 (Adv. Topics in Logic: Topics in model theory) with Professor Anand Pillay.
  • Prelim (The prelim has been passed (12/11/03), I'm pleased. Actually I'm superhappy, but don't say that to anybody...it's a secret!)
S2004
  • Math 499 (Two units of Thesis Research) I'm going to be looking for a thesis problem. Iterative differentials and characteristic p fields seem to be a good start. This, of course, is with the advice of Anand.
Sum
2004
  • Seminar organized by me. Mainly focused on discussing thesis topics.
  • Thesis. The topic is now clear. Several papers and documents have to be read before stepping forward.
F2004
  • Math 599 (Two units of Thesis Research).
  • Math 595 (Advanced topics in Logic: Model theory of valued fields) with Prof. Lou Van den Dries.
S2005
  • Math 599 (Four units of Thesis Research).
  • Isaac Newton Institute. I'll be visiting a couple of times the Isaac Newton Institute (Cambridge, UK) to assist to a few workshops which are part of the semester devoted to Model Theory and Applications to Algebra and Analysis.
  • Midwestern Graduate Logic Meeting. I'm coorganizing with Nano Tellez the 6th Annual Graduate Student Conference in Logic. I'll be held in Urbana on the weekend of April 23rd and 24th. More info, here.
Sum
2005
F2005
  • Leeds. From September to December I'll be working on my thesis in the University of Leeds (UK) supported by Marie Curie European Research Funds.
S2005-F2007
  • Barcelona. Finally, after three long years of painful distance, I've managed to move to Barcelona (thanks to the kind support of Anand) to be with Mónica while I write my thesis.

    Update: It took longer than expected but it is finally done. Thesis defense date: April 10th, 2008.

Additionally, I'm attending the following seminars and "academic" activities:
  • Logic Seminar (Tuesday).
  • Logic Lunch (Tuesday).
  • Model Theory Seminar (Followed by the traditional Logic Beer) (Friday).
  • Compulsive reading and other mental dissorders. (Daily)

Previous work (In Colombia)

My undergraduate term paper in the Universidad Nacional de Colombia, "Teoría de Modelos Geométrica y Aplicaciones" (Geometric Model Theory and Applications), was an introduction to the model theoretical proof, due to Ehud Hrushovski, of the Mordell-Lang conjecture for Function Fields. After that, with the support of the "Semillero de Logica Matematica", and the help of my undergraduate advisor, Prof. Andres Villaveces, I worked for a while in the generalization of some works of Boris Zilber on the so called "Hrushovski Structures" to the context of Abstract Elementary Classes.