SI2-SSE: Interdisciplinary Software Infrastructure for Differential Geometry, Lie Theory and their Applications

这项提议的目标是为微分几何领域以及微分几何发挥重要作用的数学、物理和工程领域进一步开发符号软件。拟议的工作将：提供用户社区要求的新功能，完成目前正在开发的一揽子程序，重新设计关键部件以提高计算效率，开发现有算法和代码的升级，其性能不能支持研究需求。具体目标包括：[1]开发用于处理抽象微分形式和齐次空间上的张量分析的新的无坐标计算环境；[2]实/复李代数的结构理论及其表示的软件；[3)实现具有对称性的张量的Young Tableaux理论以解决在大型张量计算中出现的资源和性能问题；[4]用于黎曼几何、复流形和Kahler几何以及辛几何中子流形理论的新的符号计算程序；[5]用于外部微分系统的全面的新程序包；在数学的所有核心学科中，微分几何的独特之处在于它与纯数学、应用数学、物理、工程甚至计算机科学中的许多其他学科相结合。PI的微分几何(DG)软件包为微分几何及其许多应用领域的研究和教学奠定了单一、统一的符号计算环境的基础。这项建议的目标是增加新的计算环境以满足特定的应用需求，增加使各种子包成熟的基本功能，升级具有性能限制的例程，并显著扩展DG数据库，包括李代数、群作用、可积系统和爱因斯坦方程的解。该软件的早期版本已经建立了一个重要的用户社区。社区反馈已经决定了本提案中的大部分具体计划议程。犹他州立大学和Maplesoft之间独特的合作伙伴关系确保DG软件满足扩展用户社区(具有不同级别的符号计算经验)所要求的高标准的可靠性、易用性、文档和支持以及寿命。虽然DG最初是作为研究工具设计的，但它也提供了一种在课堂上教授微分几何及其应用的创新方法。DG的所有发展都是考虑到这一点而实施的。PI将在犹他州立大学主持一个题为：微分几何、李理论和应用中的符号方法的研讨会。这个工作坊将包括实践培训课程，以及关于符号方法在微分几何问题中的应用的讲座。这个研讨会还将提供一个理想的场所来调查参与者的研究兴趣，以推动未来的代码开发。本科生和研究生的参与是这个项目的一个重要组成部分。在一般的计算机代数系统，特别是微分几何的工作中获得的经验，对学生未来的教育活动和/或未来的就业是有价值的。

The goal of this proposal is to further develop symbolic software for the field of differential geometry and those areas of mathematics, physics and engineering where differential geometry plays an essential role. The proposed work will: provide new functionalities requested by the user community, complete packages currently under development, redesign critical components for improved computational efficiency, develop upgrades of existing algorithms and code whose performance does not support the demands of research. Specific objectives include [1] the development of a new coordinate-free computational environment for work with abstract differential forms and for tensor analysis on homogeneous spaces; [2] software for the structure theory of real/complex Lie algebras and their representations; [3] implementation of the theory of Young tableaux for tensors with symmetry to address resource and performance problems arising in large tensor computations; [4] new programs for symbolic computations for sub-manifold theory in Riemannian geometry, complex manifolds and Kahler geometry, and symplectic geometry; [5] a comprehensive new package for exterior differential systems; and [6] expansion of various data-bases of Lie algebras, differential equations, and exact solutions in general relativity.Of all the core disciplines in mathematics, differential geometry is unique in that it interfaces with so many other subjects in pure mathematics, applied mathematics, physics, engineering, and even computer science. The PI's DifferentialGeometry (DG) software package has laid the foundation for a single, unified symbolic computational environment for research and teaching in differential geometry and its many application areas. The goal of this proposal is to add new computational environments to address specific application needs, to add basic functionalities that will bring various sub-packages to maturity, to upgrade routines with performance limitation, and significantly extend the DG data-bases of Lie algebras, group actions, integrable systems, and solutions of the Einstein equations. Earlier versions of this software have established a significant user community. Community feedback has dictated much of the specific program agenda in this proposal. A unique partnership between Utah State University and Maplesoft insures that the DG software meets the high standards of reliability, ease of use, documentation and support, and longevity that a extended user community (with diverse levels of symbolic computational experience) demands.While originially designed as a research tool, DG also provides an innovative approach to teaching differential geometry and its applications in the classroom. All developments in DG are implemented with this in mind.The PI will host a workshop at Utah State University entitled: Symbolic Methods in Differential Geometry, Lie Theory and Applications. This workshop will consist of hands-on training sessions, and lectures on applications of symbolic methods to problems in differential geometry. This workshop will also provide an ideal venue to survey participant research interests to drive future code development.Student involvement at the undergraduate and graduate levels is an important component of this project. The experience gained in working with computer algebra systems in general, and differential geometry in particular, is valuable to the student for future educational activities and/or future employment.