Ian Anderson - US grants

Ian Anderson will continue his investigations of the variational bicomplex. This is a topic which has rapidly moved into the forefront of research into the calculus of variations. Its applications range from conservation laws of partial differential equations to Riemannian structures and variational problems in differential geometry. The variational bicomplex is a bicomplex of differential forms which derives its name from the fact that one of its differentials may be identified with the Euler-Lagrange operator from the calculus of variations. Anderson's work will focus principally on the calculation of the cohomology of the variational bicomplex. When applied to systems of differential equations the importance of this cohomology arises from the fact that it is generated, in part, by the conservation laws of the equations. In this context Anderson will investigate the case of over-determined systems of equations. Group actions on a fiber bundle also give rise to the problem of calculating equivariant cohomology. Here Anderson will carry out the computations in cases of interest to mathematical physics.

The principal investigator will continue his research on the variational bicomplex. This theory was first introduced approximately fifteen years ago to provide a uniform approach to multi-integral problems in the calculus of variations. The bicomplex has also been used to solve inverse problems in calculus of variations. The variational bicomplex plays a role in the geometric theory of differential equations somewhat analogous to the role played by the deRham complex in finite dimensional manifolds. This theme will be developed in the research with application to specific examples a primary source of motivation. It is often useful to create a "calculus" or formalism which can be applied to a large number of problems of a class. The formalism being studied in this research applies to the class of problems known as variational problems. These problems arise naturally in a very wide range of circumstances in natural science. A simple example would be the problem of determining, among all curves on a surface connecting two points, the one with the shortest length.

9403788 Anderson The research concerns the study of the variational bicomplex and its applications to differential equations and conservation laws in mathematical physics using differential-geometric and topological techniques. Particular attention will be paid to the Euler-Lagrange complex in the setting of the modern theory of the calculus of variations. The methods are applicable to analyzing the Einstein vacuum equations. They are also applicable to to many systems of partial differential equations which arise in mathematical physics and theoretical mechanics. ***

Abstract Proposal: DMS-9804833 Principal Investigators: Ian Anderson and Mark Fels Formal differential geometry is the study of localized deformation invariants defined on the solution space of a given differential relation. These deformation invariants arise as cohomology classes in various invariant variational bicomplexes defined on the jet spaces of the given differential relation. The objectives of this research proposal are to develop new methods for the computation of these localized deformation invariants, to study the properties of these invariants through the use of jet bundle techniques, and to develop new applications of formal geometry in global analysis, integrable systems and mathematical physics. Three avenues of research are proposed. The first project will explore the relationships between the variational bicomplex for two differential equations which are related by one of the various integration or reduction methods, some of which date back to Lie and Vessiot. The second project is based upon two new extensions of the inverse problem of the calculus of variations. One extension is directed towards the existence and classification of Hamiltonian structures; the other extension seeks to characterize integral invariants arising in global analysis such as the Kazdan-Warner integrals which obstruct the solution to the prescribed curvature problem. The third project is motivated by the role that the variational bicomplex plays in the general theory of characteristic classes and seeks to generalize this role to general pseudo-group actions on fiber bundles using the method of moving frames and differential invariant theory. The prevalent theme of the proposed work is symmetry. By the symmetry of an object one means the group of motions in space which leave the object unchanged. The symmetry of a cube includes discrete rotations by 90 degrees about each one of its 3 axis. The symmetry group of a sphere includes the continuous rotations t hrough any angle about any axis through the center of the sphere. In problems in applied mathematics, differential geometry and mathematical physics, one is interested in the symmetries which leave the governing (differential) equations unchanged. Such symmetries are extremely important in studying both the analytical and qualitative features of the governing equations. Indeed, almost all known exact solutions of important non-linear equations (such as the Einstein, Yang-Mills, or harmonic map equations) have a high degree of symmetry and can be obtained by symmetry methods. Nevertheless, there are many theoretical and practical issues yet to be studied. In particular, theoretical techniques which have previously been computationally inaccessible can now be investigated using powerful symbolic mathematics computer programs.

Anderson

The investigator develops and implements algorithms for a number
of classification problems in the areas of Lie algebras,
differential geometry, and differential equations. For each of
the problems of interest, the list of representative objects is
very large. Therefore he encodes these classification results
within a computer algebra system and develops software that
allows for the efficient but flexible manipulation of the large
amounts of data contained in these classification results. This
work provides researchers with powerful new tools to find
examples and test conjectures on a wide range of subjects. The
specific classification problems that are addressed first deal
with the classification of low dimensional Lie algebras and their
sub-algebras, the classification of vector field systems in 2 and
3 dimensions, and the classification of 4- and 5-dimensional
space-times with symmetry.

The classification of mathematical structures is a central theme
in many branches of mathematics. While these classification
problems can be easy to state, the solutions are often very long
and complex and therefore not accessible to the larger
mathematical and scientific community. A major component of this
project explores new nontraditional methodologies (based on
computer algebra systems and also data-based Java applets) for
the dissemination and generalization of these results.
Applications to the automated solution of differential equations
are studied. The computer software that is being developed to
support these specific goals has applications in other areas of
mathematical research and in theoretical physics. It also has
demonstrated pedagogical value for both undergraduate and
graduate mathematics education.

This is an award for the construction of a Distributed Data Analysis for Neutron Scattering Experiments (DANSE) at the Spallation Neutron Source. It is supported by the Instrumentation for Materials Research- Mid Scale Instrumentation project program in DMR, the Office of Multidisciplinary Activity in the Mathematical and Physical Sciences Directorate, as well as the Chemistry division in DMR and the Chemical Transport Division in Engineering Directorate. The goals of the DANSE project are to build a software system that 1) enables new and more sophisticated science to be performed with neutron scattering experiments, 2) makes the analysis of data easier for all scientists, and 3) provides a robust software infrastructure that can be maintained in the future. The DANSE project was prompted by the development of the Spallation Neutron Source (http://www.sns.gov) (SNS). In 2006 the SNS will start to produce intense beams of neutrons to be used as probes of materials, molecules, and condensed matter. Neutron scattering experiments performed at the SNS will produce data of unprecedented detail on the positions and motions of atoms and spins in materials, molecules, and condensed matter. The raw experimental data acquired using the SNS instruments are not simple to interpret, and new software is required to transform the data into useful forms. Using several major advances in computational materials science that have occurred over the past decade the DANSE project will provide new data reduction and interpretation capabilities beyond what are available today. The DANSE project includes a central resource activity centered at Caltech, and 5 components based at different institutions: diffraction led by Simon J.L. Billinge of Michigan State Univ., engineering diffraction led by Ersan Ustundag of Iowa State Univ. , reflectometry led by Paul Kienzle of the Univ. of Maryland, small-angle scattering led by Paul Butler of the Univ. of Tennessee, and inelastic scattering led by Frans Trouw of Los Alamos with B. Fultz.
Information about the project is available at http://wiki.cacr.caltech.edu/danse/index.php/Main_Page
The project is helping to organize the neutron scattering science community in the U.S., and has generated worldwide interest. DANSE is a natural application for grid-based computing, and the layered design of the DANSE framework was planned for migration to the TeraGrid, or a similar future cyber infrastructure. The DANSE framework could be adapted for data analysis in other fields of science. An outreach effort has been planned as collaboration with education professionals at Iowa State University.

The principle objective of this proposal is to develop new methods for the exact solution of ordinary and partial differential equations. The research topics contained in this proposal are motivated by recent advances in symbolic methods and recent theoretical developments in the field of geometric analysis of differential equations. The theoretical developments provide for a new, unified, coherent approach to symmetry-based solution techniques. Advances in symbolic methods for symmetry computations and in the implementation of symbolic software for computations in differential geometry and Lie theory provide the computational environment for testing the practical utility of these theoretical constructions. The proposed research activities balance mathematical formalism and rigor with a desire to create effective and efficient new algorithms for solving differential equations which build upon and complement existing solution methodologies.

Almost all processes in the physical sciences and engineering are modeled by differential equations of one kind or another. By studying these differential equations and their solution, one gains understanding of how these processes evolve. Mathematicians use analytical, numerical, geometric and algebraic methods to study differential equations. This proposal deals with geometric and algebraic methods since these methods are best suited for implementation in interactive computer algebra systems (CAS). This work will expand the number and kinds of equations which can be solved by CAS. A unique feature of this proposal is the investigators ability to quickly implement new theoretical advances and distribute them to a very broad clientel of mathematicians, scientists and engineers. In addition, the computer software developed by the investigators is very useful for training advanced undergraduate and graduate students in important topics in geometry and algebra.

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.

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.