At the heart of each project in this proposal is a homological question related to combinatorial structure resulting from a group action in geometry. The first projects provide structural results via polyhedral geometry for free resolutions over a polynomial ring and a smooth toric variety. The later projects focus on hypergeometric systems; these are certain systems of linear PDEs that arise naturally from a torus action, or more generally, from a reductive group action, and are expressible through a D-module variant of Koszul homology. In each project, group actions induce algebraic gradings that contain combinatorial and geometric information. This proposal aims to isolate and exploit the induced polyhedral data structures through graded complexes from homological algebra, including free resolutions, Koszul complexes, cellular resolutions, and complexes that compute local cohomology. The projects call on methods from a broad span of mathematical areas, including homological algebra, toric geometry, representation theory, computer algebra, complex analysis, topology, and tropical geometry.
This proposal will improve diversity in the mathematical sciences and education in commutative algebra and algebraic geometry, at the undergraduate and graduate levels. Specifically, it contributes to broader impacts in three ways: mentoring undergraduate and graduate students, with special attention to female students; development of a graduate course and other opportunities for learning and dissemination; and software development to bring universal access to computational advances that result from current research. The PI has a history of commitment to these activities and is currently organizing a mentoring groups for women, compiling lectures aimed at graduate students, presenting at and organizing conferences, and developing computer algebra software.