CAREER: Problems in Commutative and Homological algebra

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

This is a project in commutative algebra, with connections to algebraic geometry, combinatorics, and arithmetic geometry. Commutative algebra is a field of abstract algebra that aims to understand the solution sets of system of polynomial equations. It is a fundamental area of research with applications and connections to fields such as robotics, statistics, and physics. This project involves studying classical systems of polynomial equations in familiar settings such as the real or complex numbers, but also the less understood setting of mixed characteristic, which has connections with number theory and arithmetic geometry. This award will also support activities in collaboration with local elementary and middle schools and graduate student training, as well as the promotion of work by early-career researchers. The PI will pursue research projects in homological and commutative algebra relating to the study and applications of p-derivations in mixed characteristic commutative algebra, symbolic powers, and cohomological support varieties. Many techniques in commutative algebra and algebraic geometry only work for an algebra over a field, because of the use of resolution of singularities and vanishing theorems in characteristic zero and the homological properties of the Frobenius map in positive characteristic. In contrast, the mixed characteristic setting is often more delicate, and many questions remain open only in that setting. Recent developments have shown that p-derivations, a tool from arithmetic geometry, can be applied to solve problems arising in mixed characteristic commutative algebra, especially when applied together with differential operators. These provide new avenues of research that will further these applications. The project will also address homological questions motivated by recent major breakthroughs related to cohomological support varieties and applications of the homotopy Lie algebra of a ring; and symbolic powers, an algebraic tool that can be used to answer classical geometric questions. This project is jointly funded by the Algebra and Number Theory Program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.