Computational Equipment for Approximation Theory, Control Theory, Graph Theory

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NSF Proposal DMS-0215442PIs: Hardin, Ellingham, Horn, Saff, and Schumaker ABSTRACTComputational Equipment for Approximation Theory, ControlTheory, and Graph TheoryThe Department of Mathematics at Vanderbilt University willpurchase computer equipment dedicated to supporting researchin the mathematical sciences. The equipment will be used forseveral research projects, as follows. Professor Ellinghamwill investigate conjectures in graph theory involvingsurface embeddings, traversability, and independent sets.Professor Hardin will construct and develop applicationsof local orthogonal bases. Professor Horn will investigatethe controllability of linked elastic structurres usingnumerical methods. Professors Hardin and Saff willinvestigate discrete minimum energy problems on the sphere.Professors Saff and Schumaker will research the theory andapplications of splines, especially on the sphere.Professor Ellingham's research involves graphs, which aremathematical models of networks. Communication,transportation and other networks are often modeled usingtechniques from graph theory. Professor Hardin's researchinvolves wavelets, which are an important tool forcompressing images, sounds, and other real-world information.They may be applied to problems in fitting equations to datapoints in any area of science, or processing images, forexample from medical scanning equipment. Professor Horn'sresearch involves control theory, which allows computationof how forces should be applied to a complicated system toachieve a desired outcome. There are applications to manyreal-world structures such as trusses, robot arms, solarpanels, and suspension bridges. The work by ProfessorsHardin and Saff involves the best way to place points ona sphere so that the sphere is covered as uniformly aspossible. This problem has applications in many fields,including the structure of viruses, crystallography,structure of molecules, and electrostatics. The researchby Professors Saff and Schumaker involves trying to provideaccurate but easily computable representations of quantitieson the surface of a sphere. For example, these technique maybe used to construct an accurate model of the Earth'sgravitational field.