SI2- SSE: Symbolic Toolboxes for Differential Geometry and Mathematical Physics

SI2-SSE：微分几何和数学物理的符号工具箱

• 批准号: 1642404
• 负责人: Ian Anderson
• 金额: $ 31.42万
• 依托单位: Utah State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-10-01 至 2021-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1642404&HistoricalAwards=false
• 关键词: SI2; SSE; Symbolic; Toolboxes; Differential

This project develops the DifferentialGeometry (DG) software for research and educational use across a broad spectrum of disciplines, from mathematics to physics and engineering. With this software many pencil and paper calculations in differential geometry and its applications, calculations which were previously intractable, can now be performed quickly, reliably, and with relative ease. DG provides extensive mathematical infrastructure which supports the formulation of new conjectures, the creation of examples and application of theoretical results, the ability to easily verify many results in the existing scientific literature, and the ability to effortlessly share complex calculations with collaborators, colleagues, and students. DG libraries also provide access - for both experts and non-experts - to large tracts of scientific and mathematical knowledge. A number of undergraduate and graduate students will participate in this project, performing software development and exploring applications of DG to research problems in mathematics and physics. In particular, DG provides an excellent means to get undergraduates involved in advanced research projects which normally would be accessible only to graduate students.This project creates symbolic computational toolboxes and libraries to support research needs in differential geometry, relativity and field theory, differential equations and integrable systems, and Lie theory. These toolboxes and libraries will provide new infrastructure for symbolic computing in differential geometry and its applications; meet specific user community demands; and explore new areas where symbolic methods have heretofore been unused. Project highlights include new objects and environments for working with submanifolds, general connections, differential operators, and constrained jet spaces. Tools for analyzing asymptotic structure of spacetimes represent an innovative use of computer algebra. A new toolbox will be created which incorporates much of the extensive mathematical literature on the classification of Lie subalgebras. This project will provide, for the first time, a comprehensive symbolic toolkit for investigations of integrable Partial Differential Equations (PDE). New libraries of symbolic data include symmetric and isotropy irreducible homogeneous spaces, solutions of relativistic field equations and their properties, integrable PDE and their properties. As libraries of symbolic data are created, DG is used to validate and correct results in the literature. Software development and community engagement projects which will ensure sustainability are included.

该项目开发微分几何（DG）软件，用于从数学到物理和工程等广泛学科的研究和教育用途。有了这个软件，许多铅笔和纸计算微分几何及其应用，计算这是以前棘手的，现在可以快速，可靠地执行，并相对容易。 DG提供广泛的数学基础设施，支持新猜想的提出、示例的创建和理论结果的应用、轻松验证现有科学文献中许多结果的能力，以及与合作者、同事和学生轻松分享复杂计算的能力。DG图书馆还为专家和非专家提供了大量的科学和数学知识。一些本科生和研究生将参与这个项目，进行软件开发和探索DG的应用，以研究数学和物理问题。 特别是，DG提供了一个很好的手段，让本科生参与先进的研究项目，通常只有研究生才能访问。这个项目创建了符号计算工具箱和图书馆，以支持微分几何，相对论和场论，微分方程和可积系统，李理论的研究需求。这些工具箱和库将为微分几何及其应用中的符号计算提供新的基础设施;满足特定用户社区的需求;并探索符号方法迄今未使用的新领域。项目亮点包括新的对象和环境，用于处理子流形、一般连接、微分算子和约束喷射空间。分析时空渐近结构的工具代表了计算机代数的创新应用。一个新的工具箱将被创建，其中包括大量的广泛的数学文献分类的李子代数。该项目将首次为可积偏微分方程（PDE）的研究提供一个全面的符号工具包。新的符号数据库包括对称和各向同性的不可约齐性空间，相对论场方程的解及其性质，可积偏微分方程及其性质。作为图书馆的符号数据的创建，DG是用来验证和纠正文献中的结果。软件开发和社区参与项目将确保可持续性。

项目成果

Rainich conditions in (2 + 1)-dimensional gravity

(2 1) 维引力中的雷尼奇条件

• DOI: 10.1063/1.4974091
• 发表时间: 2017
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Krongos, D. S.;Torre, C. G.
• 通讯作者: Torre, C. G.

Explicit ambient metrics and holonomy

明确的环境指标和完整度

• DOI: 10.4310/jdg/1580526015
• 发表时间: 2020
• 期刊: Journal of Differential Geometry
• 影响因子: 2.5
• 作者: Anderson, Ian M.;Leistner, Thomas;Nurowski, Paweł
• 通讯作者: Nurowski, Paweł

Spacetime groups

时空群

• DOI: 10.1063/5.0001460
• 发表时间: 2020
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Anderson, Ian;Torre, Charles
• 通讯作者: Torre, Charles

Affine symmetry, geodesics, and homogeneous spacetimes

仿射对称、测地线和齐次时空

• DOI: 10.1007/s10714-018-2422-0
• 发表时间: 2018
• 期刊: General Relativity and Gravitation
• 影响因子: 2.8
• 作者: Maughan, David;Torre, Charles
• 通讯作者: Torre, Charles