Research

Disclaimer: This page is severely outdated. For an overview of my research, please visit my Papers section. One day soon, I will have the time to write something coherent and current about my research interests.

My main area of specialization is Computational Commutative Algebra and algebraic geometry with emphasis on their applications. I have done research in the following subjects:

• Algebraic Statistics: This area proposes computational commutative algebra as a tool in experimental design, discrete probability, and statistical modeling. In my Ph. D. thesis, I studied a particular statistical model known as Bayesian network from the point of view of computational algebraic geometry. My contributions to this subject are:
  • (with Marta Casanellas and Seth Sullivant). Catalog of small phylogenetic trees. In Algebraic Statistics for Computational Biology (editors L. Pachter and B. Sturmfels), Cambridge University Press, 2005, pp. 347-358.
  • Algebraic statistics in model selection. In Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence (editors M. Chickering and J. Halpern), 2004, pp. 177-184.
  • (with Mike Stillman and Bernd Sturmfels). Algebraic geometry for Bayesian networks. In Journal of Symbolic Computation 39/3-4 (2005), pp. 331-355.
  • Algebraic geometry of Bayesian networks. Ph. D. thesis (2004). Department of Mathematics. Virginia Polytechnic Institute and State University.
  • Polynomial constraints of Bayesian networks with hidden variables. Manuscript.
  • (with Mathias Drton). A comparison of a Gaussian and a binary ancestral graph model. Manuscript.

  You can check the following links to know more about algebraic statistics:

  • Algebraic Statistics for Computational Biology.
  • Clay Mathematics Institute: Workshop on Algebraic Statistics and Computational Biology.
  • ARCC Workshop: Computational Algebraic Statistics.
  • Algebraic Statistics: Theory and Practice.
  • John von Neumann Lectures 2003.

  ---

• Algebraic Combinatorics: I am interested in the fascinating interplay between combinatorics and commutative algebra. In particular, I am very interested in monomial ideals and toric ideals.

  I have worked on cellular resolutions of monomial ideals. To know more about the subject, you can check the book Combinatorial Commutative Algebra written by Ezra Miller and Bernd Sturmfels. I am currently working with Mike Falk on certain resolutions of monomial ideals arising from hyperplane arrangements. I gave a talk that contains some of the results in our work.

  • (with Mike Falk). Minimal Cohen-Macaulay deformations of matroid ideals. In progress.

  ---

• Toric Ideals

  ---

• Gröbner bases
  • (with Maria Alicia Avino-Diaz). Computing the additive structure of indecomposable modules over Dedekind-like rings using Gröbner bases. In Journal of Algebra. Accepted 2006.

  • Bases de Gröbner asociadas a modulos finitos (Spanish). In Miscelanea Matematica (MMS) 30 (2000), 65-70.

  ---

• Primary Decomposition: During the Fall 2002, I was a "visiting student" in the Mathematics Department at the University of Genova in Italy. Everything started having dinner at Olivier's restaurant in New Orleans with Lorenzo Robbiano in September of 2001. Then he invited me to work with the wonderful CoCoA team. We are in the process of implementing a library to perform primary decomposition of polynomial ideals in the new version of CoCoA.

  ---

• Sequential Dynamical Systems
  • (with Abdul Jarrah and Reinhard Laubenbacher). Sequential dynamical systems over words. In Applied Mathematics and Computation 174/1 (2005), pp. 500-510.
  • (with Abdul Jarrah and Reinhard Laubenbacher). Sequential Dynamical Systems. Book in progress.