Homological Commutative Algebra and Group Actions in Geometry

几何中的同调交换代数和群作用

基本信息

批准号：
1661962
负责人：
Christine Berkesch
金额：
$ 16.15万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1661962&HistoricalAwards=false
关键词：
Homological Commutative Algebra Group Actions

项目摘要

This project concerns types of symmetry in algebraic structures which arise in a wide range of areas of application, including biology, chemistry, geometry, optimization, and physics. The research involves the development of a host of new tools to express geometric properties algebraically. The education and broader impacts portion of the project integrates with the research through work with graduate students. The PI will also continue involvement in several mentoring initiatives, including the Enhancing Diversity in Graduate Education (EDGE) Program, Math Alliance, and a professional development seminar for mathematics graduate students in conjunction with advising Minnesota's Women in Math program; organization of regional and international conferences organization; and software development and distribution for the open source computer algebra system Macaulay2. The research components of this project seek to establish a foundational framework for each of the following: (1) complexes corresponding to line bundle resolutions of sheaves on smooth toric varieties, (2) free resolutions of equivariant ideals in an infinite setting, for instance, the case of a symmetric group action on a countably infinite-dimensional space, and (3) a D-module variant of Koszul homology over spherical varieties. The specific parts include (respectively): (1) developing and applying analogues for smooth toric varieties of foundational homological results for projective space, (2) determining how to track and compute invariant syzygies in infinite settings with a suitable monoid or group action, and (3) generalizing a homological framework for hypergeometric systems of PDEs from the setting of a torus action to that of a reductive group. The project will yield new sets of tools for shedding light on the underlying geometry and group actions present, by aiding in the computation of important algebro-geometric and PDE invariants.

该项目涉及代数结构中对称性的类型，这些对称性在广泛的应用领域（包括生物学，化学，几何，优化和物理学）中产生。该研究涉及开发许多新工具以以代数表达几何特性。通过与研究生的工作，教育和更广泛的影响与研究的一部分与研究融为一体。 PI还将继续参与多项指导举措，包括增强研究生教育（EDGE）计划，数学联盟的多样性，以及针对数学研究生的专业发展研讨会，并结合为明尼苏达州的数学女性提供咨询服务；区域和国际会议组织的组织；以及开源计算机代数系统MacAulay2的软件开发和分发。该项目的研究组成部分旨在为以下每个过程建立一个基础框架：（1）与平滑的圆环品种上的系束线束分辨率相对应的复合物，（2）在无限环境中免费分辨率分辨率，例如，在无限环境中的自由分辨率。对称群体对无限二维空间的对称组作用的情况，（3）Koszul同源性在球形品种上的D模块变体。特定零件包括（分别）：（1）开发和应用类似物，以使投影空间的基础同源结果的平滑曲折品种，（2）确定如何在无限设置中使用合适的单型或组动作跟踪和计算不变的syzygies，以及（3）从圆环动作的设置到还原群的PDE的高几何系统的同源框架。该项目将通过协助计算重要的代数几何和PDE不变性来生产新的工具集，以阐明存在的基础几何和群体动作。