Research Projects

The project aims to explore the relationship between the ordinary and symbolic powers of squarefree monomial ideals, which have significant applications in commutative algebra, combinatorics, and algebraic geometry. It is focusing on the Conforti-Cornuejols conjecture and Huneke's question, which are two significant challenges in this field. The conjecture states that the ordinary and symbolic powers of a squarefree monomial ideal are equal iff the ideal satisfies the packing property. Huneke's question questions whether the equality of the ordinary and symbolic powers holds from a certain point q₀ onwards, assuming they are equal up to q₀. The project seeks to identify new structural properties of optimization problems represented as squarefree monomial ideals, which hold the promise of innovative discoveries in integer programming methods. The project relies on the use of hypergraphs and edge ideals, which are used in algebraic geometry and commutative algebra to study combinatorial structures. It focuses on a specific class of edge ideals of hypergraphs and path ideals of graphs, which are generated by the set of monomials corresponding to the edges of the hypergraph. The project aims to analyze specific hypergraphs and path ideals where the conjecture holds and give the bounds for the q₀ in terms of combinatorics. The goal is to identify the common properties that lead to the equality of ordinary and symbolic powers for these specific classes of ideals, with the aim of gaining insights into the conditions for the conjecture to hold generally.