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: 1440537
• 负责人: Christine Berkesch
• 金额: $ 10.4万
• 依托单位: University of Minnesota-Twin Cities
• 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 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1440537&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.

该提案中每个项目的核心是与几何作用的组作用产生的组合结构有关的同源问题。第一个项目通过在多项式环上的自由分辨率和光滑的感谢您的多种分辨率来提供结构性结果。后来的项目着重于高几幅系统；这些是某些线性PDE的系统，它们是由圆环动作自然产生的，或更普遍地是由于还原性的动作而产生的，并且可以通过Koszul同源性的D模块变体表达。在每个项目中，小组行动都会引起包含组合和几何信息的代数等级。该建议旨在通过来自同源代数的分级复合物（包括自由分辨率，Koszul复合物，细胞分辨率和计算局部共同体学的复合物）来隔离和利用诱导的多面数据结构。这些项目呼吁从广泛的数学领域中的方法，包括同源代数，折磨几何，代表理论，计算机代数，复杂分析，拓扑结构和热带几何形状。该提案将改善数学科学和教育程度的多样性，在通勤代数和代数的几何学上，在不熟悉和研究生级的级别上。具体来说，它以三种方式促进了更广泛的影响：指导本科生和研究生，并特别关注女学生；发展研究生课程以及其他学习和传播机会；和软件开发，使人们可以普遍访问当前研究所产生的计算进步。 PI具有对这些活动的承诺历史，目前正在为女性组织一个指导小组，编写针对研究生，在和组织会议的讲座以及开发计算机代数软件。