Scaling up Mathematical Computations

在这个项目中，研究人员将进行大规模的计算，以支持数学的三个基本分支的研究：纯数学、应用数学和统计学。将这些项目联系在一起的一件事是，每一项研究计算都可以按比例进行。这种可扩展性很重要，因为这三家公司都将利用佐治亚理工学院新建立的校园计算资源，该资源旨在作为国家超级计算设施(如TeraGrid)的垫脚石。因此，该项目有可能提高TeraGrid的可见度，使其成为超越传统科学计算领域的更广泛数学研究的资源。所有这些项目都将伴随着本科生和研究生在计算方法和基本数学概念方面的培训。此外，所有电子产品(如软件包、数据集等)从这项工作开发出来的将被公开分享。第一个项目将需要与纽结理论相关的大量计算。我们将为有20个交叉点的纽结计算各种各样的纽结不变量，并为有15个交叉点的纽结计算一个广泛的彩色琼斯多项式表。这些不变量将被加载到一个精心构建的数据库中，以进行有效的搜索、猜想测试和实验。纽结数据库将是整个研究社区的宝贵资源，包括拓扑学家、几何学家，甚至分子生物学家。在应用数学的第二个项目中，我们将运行无粘流体中柔性体的演化代码。这是一种用于研究鱼类、鸟类和流体中其他相互作用体的模型。我们希望填补我们对柔性体如何与流动的流体相互作用的基本理解中的一些重大空白。统计学的第三个项目将集中在由随机微分方程定义的随机过程。由于这类过程的复杂性和模型观测结构，其统计推断面临着重大挑战。我们将使用现代非参数统计推断方法，这可能是非常计算密集的，以建立一个坚实的框架，并提高我们对这些过程的理解。这项工作的应用包括用几百种证券组成的更现实的投资组合来建模和预测投资组合风险。

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 realisticportfolio of a few hundred securities.