Computational Equipment for Approximation Theory, Control Theory, Graph Theory

Vanderbilt University, Nashville TN

Investigators

Abstract

NSF Proposal DMS-0215442 PIs: Hardin, Ellingham, Horn, Saff, and Schumaker ABSTRACT Computational Equipment for Approximation Theory, Control Theory, and Graph Theory The Department of Mathematics at Vanderbilt University will purchase computer equipment dedicated to supporting research in the mathematical sciences. The equipment will be used for several research projects, as follows. Professor Ellingham will investigate conjectures in graph theory involving surface embeddings, traversability, and independent sets. Professor Hardin will construct and develop applications of local orthogonal bases. Professor Horn will investigate the controllability of linked elastic structurres using numerical methods. Professors Hardin and Saff will investigate discrete minimum energy problems on the sphere. Professors Saff and Schumaker will research the theory and applications of splines, especially on the sphere. Professor Ellingham's research involves graphs, which are mathematical models of networks. Communication, transportation and other networks are often modeled using techniques from graph theory. Professor Hardin's research involves wavelets, which are an important tool for compressing images, sounds, and other real-world information. They may be applied to problems in fitting equations to data points in any area of science, or processing images, for example from medical scanning equipment. Professor Horn's research involves control theory, which allows computation of how forces should be applied to a complicated system to achieve a desired outcome. There are applications to many real-world structures such as trusses, robot arms, solar panels, and suspension bridges. The work by Professors Hardin and Saff involves the best way to place points on a sphere so that the sphere is covered as uniformly as possible. This problem has applications in many fields, including the structure of viruses, crystallography, structure of molecules, and electrostatics. The research by Professors Saff and Schumaker involves trying to provide accurate but easily computable representations of quantities on the surface of a sphere. For example, these technique may be used to construct an accurate model of the Earth's gravitational field.