LEAPS-MPS: Homological investigations of differential operators, almost complete intersections, and Gorenstein ideals

Oberlin College, Oberlin OH

Investigators

Abstract

Commutative algebra is an essential branch of mathematics that investigates the solutions of systems of polynomial equations, which can be represented as curves, surfaces, or higher dimensional spaces. It aims to discover geometric properties of these solution sets from a purely algebraic perspective by examining important algebraic structures called commutative rings and modules that naturally encode the desired geometric information. This project will investigate several such algebraic structures, including differential operators, which can be used to detect and quantify singularities (i.e., measure how smooth or pointy a curve or surface is). Such geometric data has potential far-reaching applications in the sciences and modern technologies including medical imaging and robotics. Central to this project is the objective to broaden participation of underrepresented groups in mathematics through mentored undergraduate research projects, professional development opportunities, and preparation for STEM careers. More specifically, this project focuses on the development of new homological techniques to answer fundamental questions about several classical objects in commutative algebra which remain at the heart of modern research in the field, namely, differential operators, almost complete intersections, and Gorenstein ideals, as well as ideals that have underlying combinatorial structures. Each investigation will involve understanding, and in several cases constructing, minimal free resolutions as primary homological tools for understanding these algebraic objects. The investigation will proceed in four main directions: (1) construction of free resolutions of the modules of differential operators over certain hypersurface rings, (2) calculation of Poincare series over almost complete intersection rings, (3) investigation of Koszul homology and other homological invariants of certain Gorenstein rings and ideals, and (4) investigation of the homological properties of certain edge ideals and their free resolutions. 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.