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

该项目开发了微分几何（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，许多在微分几何及其应用中的铅笔和纸计算，以前难以处理的计算，现在可以快速，可靠，相对容易地执行。DG提供了广泛的数学基础设施，支持新猜想的制定、例子的创建和理论结果的应用、轻松验证现有科学文献中的许多结果的能力，以及与合作者、同事和学生轻松共享复杂计算的能力。DG图书馆还为专家和非专家提供了获取大量科学和数学知识的途径。一些本科生和研究生将参与该项目，进行软件开发和探索DG在数学和物理问题研究中的应用。特别是，DG提供了一个很好的途径，让本科生参与到通常只有研究生才能参与的高级研究项目中来。该项目创建符号计算工具箱和库，以支持微分几何、相对论和场论、微分方程和可积系统以及李理论的研究需求。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；满足特定的用户群体需求；探索符号方法尚未使用的新领域。项目重点包括用于处理子流形、一般连接、微分算子和受限射流空间的新对象和环境。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，其中包含了大量关于李子代数分类的数学文献。该项目将首次为研究可积偏微分方程（PDE）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不可约齐次空间、相对论场方程的解及其性质、可积偏微分方程及其性质。随着符号数据库的创建，DG用于验证和纠正文献中的结果。包括确保可持续性的软件开发和社区参与项目。

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.