Multigraded commutative algebra and asymptotic behavior of filtrations of ideals

Arizona State University, Scottsdale AZ

Investigators

Abstract

This project focuses on several problems in commutative algebra, the branch of mathematics that explores properties of polynomial equations, which are fundamental for modeling diverse phenomena in science and engineering. As a result, commutative algebra has strong connections with biology, computer science, physics, and other quantitative fields. When equations involve multiple variables, their comprehensive study can become intractable. A powerful strategy in such cases involves decomposing polynomials into smaller pieces and using information from these components to derive general properties, a theme known as multigraded commutative algebra. Another significant approach concerns understanding the asymptotic behavior of sequences of sets of equations known as filtrations. This project will advance these research directions by addressing key questions within the field. Furthermore, this project will have a broader impact on the postdoctoral, graduate, and undergraduate student population through mentoring initiatives and the organization of seminars, conferences, and workshops. The project will advance the understanding of Hilbert series through a detailed investigation of multidegree support and K-polynomials of multiprojective schemes. This research will explore connections between the topology of schemes and the combinatorial aspects of K-polynomials, with direct implications for Schubert geometry, toric geometry, and multiparameter persistent homology. Additionally, the project will employ Presburger and Ehrhart methods to analyze the quasi-polynomial behavior of homological functors applied to multigraded modules. Divisorial filtrations, which are defined via valuations, exhibit intricate geometric properties and include significant examples such as symbolic powers and integral closure powers of ideals. The project will study the growth rate of the number of generators of these filtrations. Furthermore, the project will investigate whether divisorial filtrations are F-split, potentially indicating mild F-singularities in their blowup algebras and low complexities in the growth of homological functors. 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.