Homological commutative algebra, polyhedral structure, and algebraic geometry

同调交换代数、多面体结构和代数几何

• 批准号: 1303083
• 负责人: Christine Berkesch
• 金额: $ 10.4万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303083&HistoricalAwards=false
• 关键词: Homological; commutative; algebra; polyhedral; structure

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.

在这个建议中的每个项目的核心是一个同调问题，涉及到组合结构，从几何中的群作用。第一个项目提供了结构的结果，通过多面体几何自由决议的多项式环和光滑复曲面品种。后来的项目专注于超几何系统;这些是某些线性偏微分方程系统，它们自然地产生于环面作用，或者更一般地，从还原群作用，并且可以通过Koszul同调的D-模变体来表达。在每一个项目中，小组行动诱导代数评分，包含组合和几何信息。该提案旨在通过同调代数的分次复体来隔离和利用诱导的多面体数据结构，包括自由分辨率，Koszul复体，细胞分辨率和计算局部上同调的复体。这些项目需要来自广泛数学领域的方法，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。这一建议将提高本科和研究生数学科学和交换代数和代数几何教育的多样性。具体而言，它以三种方式促进更广泛的影响：指导本科生和研究生，特别关注女生;开发研究生课程和其他学习和传播机会;开发软件，普及当前研究所取得的计算进步。公共关系协会历来致力于这些活动，目前正在组织妇女辅导小组，编写针对研究生的讲座，在会议上发言和组织会议，并开发计算机代数软件。