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

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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.