"SCREMS" Computational Mathematics Research at UTM

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Investigators from the Department of Mathematics at The University of Texas at Austin are involved in projects that include the development of deterministic high order numerical schemes for Boltzmann type kinetic transport models and the study of non-equilibrium statistical mechanics; computer-assisted theorem proving in analysis related to the onset of instabilities in dynamical systems; development of computational tools for studying the material and conformational properties of DNA at various scales; computations of random fronts in two or three space dimensional stochastic reaction-diffusion equations and generation of a large solution sample space to analyze front statistics; computational investigations for large-scale distribution of predicted orders of the Tate-Shafarevich groups of the quadratic twists of a fixed elliptic curve; and the growth of the minimal weight in various families of error correcting codes. Each of these projects shares the need for large amounts of uninterrupted computation cycles. A single computer-assisted proof in the analysis of dynamical systems can consume over 3 years of CPU time on a 1GHz Pentium class processor. The computational requirements of testing numerical methods for kinetic transport models are essentially open ended; investigators currently use all available resources continuously. Similarly, the computational requirements of investigating the large-scale distribution of orders of Tate-Shafarevich groups are practically unlimited. The scope and the magnitude of the projects outlined above warrant the deployment of a substantial dedicated computational facility for the pursuit of these objectives.The NSF funds awarded for the purchase of a computational server cluster provide considerable momentum to the vigorous pursuit of the research agenda outlined in preceding paragraph. All of these projects deal with problems that are of current interest in computational mathematics. They are part of an effort to understand the mechanisms behind phenomena that play an important role in solid state physics, biology, cryptography, and other sciences. In addition, they advance the state of the art in computational mathematics, which brings new problems within the reach of mathematical analysis. Some of the projects also involve interactions with software developers, which contributes to the design of new software for scientific applications. The applications of this research are wide ranging and profound. They include the modeling of nano-electronic and biological structures; analyzing the stability of dynamical systems such as plasma confinement in fusion energy accelerators, celestial mechanics, and models for atmospheric dynamics; understanding how forest fires and turbulent fire fronts propagate; and investigating computer security and quantum computing. The results are communicated to as wide an audience as possible, through discussions, lectures, refereed journals, preprint archives, web sites, etc. The research involves the participation and training of graduate and undergraduate students.