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.
