Multigraded structures in Algebra, Geometry, and Combinatorics

This award supports research in commutative algebra and algebraic geometry, with strong connections to combinatorics and singularity theory. Commutative algebra provides the local algebraic framework for algebraic geometry—a central area of modern mathematics concerned with understanding the geometric structure of solution sets to systems of polynomial equations. These solution sets, known as algebraic varieties, appear throughout mathematics and are often challenging to study due to their intricate geometric and algebraic features. To address these challenges, the principal investigator will investigate the rich multigraded structures that naturally arise in many such varieties. This research will generate new insights at the intersection of algebra, geometry, and combinatorics while contributing to the training and development of the next generation of mathematical scientists. The project also includes mentoring and training of graduate students, as well as fostering a learning community at the principal investigator’s home institution for those interested in related research topics.

The research will center on four main directions, each guided by the unifying presence of graded or multigraded structures. The first project aims to develop new formulas in elimination theory, compute the defining equations of certain blowup algebras, and establish novel criteria for detecting integral dependence. These formulas and computations will have important tangible applications in singularity theory and applied areas like geometric modeling. The second project aims to study the multidegree and K-polynomials of multiprojective varieties—for example, by investigating the saturated Newton polytope and Lorentzian properties. This work will advance the understanding of combinatorially-defined polynomials, such as skew Schur, Schubert, and Grothendieck polynomials. The third project aims to investigate key structural properties of the Hilbert scheme of a multiprojective space. The guiding theme is to understand the closed subschemes of a multiprojective space and their possible deformations. The fourth project aims to describe nonreduced scheme structures using Noetherian differential operators. This approach will provide innovative applications of differential operators in commutative algebra and algebraic geometry.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Award Number: 2502321
Principal Investigator: Yairon Cid-Ruiz
Funds Obligated: $119,093
State: NC