Research

"In theory there is no difference
beetween theory and practice.
In practice there is."

Much of my research is motivated by the combinatorics that underlie many mathematical objects, which function as a skeleton of it and, in many cases, determine some of its properties. In general, I am interested in the interactions of three areas of mathematics: combinatorics, algebra, and optimization. I have been working in:

  • Algebraic Combinatorics and Commutative Algebra.
  • Combinatorics and Polyhedral Geometry.
  • Combinatorial Optimization and Linear Programming.

Currently, my research work focuses mainly on the next two topics:

  • Minimal free resolutions of monomial ideals, and
  • Arithmetical structures of matrices.

The problem of finding a minimal free resolution in a non-recursive form of a monomial ideal was proposed by Irving Kaplansky in the early 1960s. Since then, it has been a central problem of combinatorial commutative algebra. Together with my students, we are interested in finding minimal free resolution of a monomial ideal in terms of some combinatorial substructures of the ideal, which we call a combinatorial resolution. We are also working on an invariant of the monomial ideal, which we call the signature of the ideal.

Arithmetical structures of graphs arise in the study of degenerating curves in algebraic geometry as some intersection matrices. After that, its study was extended to square non-negative integer matrices.
The arithmetical structures of a matrix can be thought of as a multivariate version of the classical concept of an eigenvalue-eigenvector pair in linear algebra, which is also a subset of the solutions of the Diophantine equation defined by the determinant of a matrix with variables on the diagonal that are stable in the sense that there are no other solutions above this one. Together with my former Ph.D. student Ralihe Villagran, we proved that there is an algorithm to calculate the structures of a matrix and, more generally, of a class of polynomials that we called dominated.

You can find a complete list of my publications on my Google Scholar page.

In the past, I was also interested in:

  • The study of a graph critical ideals
  • Applications for allocation problems in bipartite graphs.

The critical ideals of a graph are determinantal ideals of the Laplacian matrix generalized of a graph. Critical Ideals generalize the characteristic polynomial of a graph and the concept of critical group. The critical group of a graph has been studied in different areas of mathematics. In algebraic geometry is known as the group of components, Picard group or as the Jacobian. In statistical mechanics, is known as the abelian model of sandpiles. And in Combinatorics is known as the group of sandpiles or the game of chip-firing.

On the other hand, allocation problem in bipartite graphs is one of the most studied problems in Combinatorial Optimization, and one of which has more potential applications. This part of my work was done in conjunction with the group of Cipriano Santos in the Hewlett-Packard Laboratories in Palo Alto, CA, and now we have 5 international patents.