Scaling up Mathematical Computations

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

In this project, the investigators will perform large scale computations to support research in three fundamental branches of mathematics: pure mathematics, applied mathematics, and statistics. One thing that ties these projects together is that each of the research computations can be scaled. This scalability is important, since all three will be leveraging a newly established campus computing resource at Georgia Tech which is designed to serve as a stepping stone to national supercomputing facilities such as the TeraGrid. Hence, there is the potential for this project to raise the visibility of the TeraGrid as a resource for broader mathematical research beyond the traditional areas of scientific computing. Undergraduate and graduate student training in both the computational methods and the underlying mathematical concepts will accompany all of these projects. Moreover, all electronic artifacts(e.g. software packages, datasets, etc.) developed from this work will be shared publicly. The first project will require substantial computations related to knot theory. We will compute a wide variety of knot invariants for knots with 20 crossings, and an extensive table of colored Jones polynomials for knots with 15 crossings. These invariants will be loaded into a carefully structured database for efficient searching, testing of conjectures, and experimentation. The knot database will be a valuable resource to an entire community of researchers, including topologists, geometers, and even molecular biologists. For the second project in applied mathematics, we will run codes for the evolution of flexible bodies in inviscid fluids. This is a model used in the study of schooling fish, bird formations, and other interacting bodies in fluids. We hope to fill in some of the large gaps in our fundamental understanding of how flexible bodies interact with flowing fluids. The third project in statistics will focus on a stochastic processes defined by stochastic differential equations. The statistical inference for such processes faces major challenges due to their complexity and model observation structure. We will employ modern nonparametric statistical inference methods, which can be very computationally intensive, to build a solid framework and improve our understanding of these processes. Applications of this work includes modeling and forecasting portfolio risk with a more realistic portfolio of a few hundred securities.