Broadly speaking, my mathematical interests are in Commutative Algebra and its interactions with Algebraic Geometry and Combinatorics. Here is a concise list of problems/research areas:
- Structure theorems for special classes of ideals: 5 of my papers and 2 of my projects in preparation are dedicated to the structure of special classes of ideals, and their numerical characters (Betti numbers, multiplicity, projective dimension, …)
- Projective dimension and Stillman’s conjecture: 6 of my papers are dedicated either to the problem of finding sharp upper bounds in Stillman’s conjecture, or development of tools to attack this problem.
- Combinatorial Commutative Algebra: 3 of my papers formulas or bounds for projective dimension, regularity and arithmetical rank of monomial ideals.
- Symbolic powers of ideals, and ideals of points in n-dimensional projective spaces: so far I have written one paper on it, and more may be coming…
- Koszul (commutative) algebras: I have a paper on this very interesting topic, but I suspect that I may be working more in this direction.
- Conormal modules and infinite resolutions: currently one of my papers is dedicated to the conormal module and another one is on its way..
- Linkage, residual intersections, j-multiplicity, and Sally’s conjecture: although I have used linkage in most papers that I have worked on, I have realized that so far I have only written one paper on Liaison Theory. I have written also one paper on j-multiplicity and Sally’s conjecture.
Tools or topics that I like to work with include:
- Liaison theory,
- Hilbert functions,
- minimal free resolutions and homological invariants: syzygies, projective dimension, arithmetical rank, infinite resolutions,…
- symbolic (and regular) powers of ideals,
- points in projective space,
- Koszul algebras,
- combinatorial structures associated to (square-free) monomial ideals,
- residual intersections and j-multiplicity.
For a fairly updated list of research papers I have authored, see my papers.
At the link, you will find my CV.