Homological commutative algebra, polyhedral structure, and algebraic geometry

在这个提案中，每个项目的核心都是一个与几何中的群作用产生的组合结构相关的同调问题。第一个项目通过多面体几何提供了在多项式环和光滑环面簇上的自由分辨率的结构结果。后面的项目集中于超几何系统；这些是某些线性偏微分方程组的系统，它们自然地产生于环面作用，或者更一般地，来自约化的群作用，并且可以通过Koszul同调的D-模变体来表示。在每个项目中，群体行为导致包含组合和几何信息的代数分级。该建议旨在通过同调代数中的分级复形来分离和利用诱导的多面体数据结构，包括自由分辨率、Koszul复形、胞元分辨率和计算局部上同调的复形。这些项目涉及到广泛的数学领域的方法，包括同调代数、环面几何、表示论、计算机代数、复分析、拓扑学和热带几何。这项建议将提高数学科学的多样性，以及本科生和研究生水平的交换代数和代数几何教育。具体地说，它在三个方面促进了更广泛的影响：指导本科生和研究生，特别关注女学生；开发研究生课程和其他学习和传播机会；以及开发软件，使人们普遍获得当前研究的计算成果。国际和平协会一直致力于这些活动，目前正在为妇女组织一个辅导小组，编写针对研究生的讲座，在会议上发表演讲并组织会议，以及开发计算机代数软件。

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.