Homological commutative algebra, polyhedral structure, and algebraic geometry

本提案中每个项目的核心是一个与几何中的群体作用产生的组合结构相关的同源问题。第一个项目通过多面体几何在多项式环和光滑环上提供自由分辨率的结构结果。后期的项目侧重于超几何系统；这些是由环面作用自然产生的线性偏微分方程的某些系统，或者更一般地说，是由约化群作用产生的，并且可以通过科祖尔同调的d模变体来表示。在每个项目中，群体行为诱导包含组合和几何信息的代数分级。本文旨在通过同调代数的梯度复合体分离和利用诱导多面体数据结构，包括自由分辨率、Koszul复合体、细胞分辨率和计算局部上同调的复合体。这些项目需要广泛的数学领域的方法，包括同调代数、环面几何、表示理论、计算机代数、复杂分析、拓扑和热带几何。这一建议将提高数学科学的多样性，并在交换代数和代数几何教育，在本科和研究生水平。具体而言，它在三个方面有助于产生更广泛的影响：指导本科生和研究生，特别关注女学生；开设研究生课程并提供其他学习和传播的机会；以及软件开发，使人们能够普遍使用当前研究成果所带来的计算进步。PI一直致力于这些活动，目前正在组织一个妇女指导小组，编写针对研究生的讲座，在会议上发言和组织会议，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.