Permanent Structures Across Scales: Assessing Stability the Effects of Perturbations

提议：DMS-0410267PI：Christopher K.Jones机构：北卡罗来纳大学教堂山标题：跨尺度的永久结构：评估稳定性和扰动的影响摘要：跨越尺度的结构分析对数学提出了挑战，近年来产生了新的想法和理论。在这项工作中的调查者集中在与具有不同尺度的对比结构的永久结构的稳定性相关的问题上。它们在给定系统中的存在是非线性和空间物理效应之间平衡的结果，例如色散(例如，对于光通信中的孤立波)、扩散(例如，在生物介质中传播的波)或对流(例如，材料或气体中的冲击)。稳定性问题需要深入的分析，以揭示给定系统内部工作的更复杂特征。这个项目的工作将解决四个问题：(1)通过不同的空间物理效应在不同尺度上保持的波的稳定性的确定；(2)在具有不同尺度上的子结构的多维模式中的稳定性的评估；(3)宏观尺度上的稳定性对控制较小尺度上的变化的影响；以及(4)对于复杂结构，测量影响范围的不稳定表现的最佳方式。物理或生物模型中的关键问题通常涉及永久结构的稳定性，例如非线性波或模式。这个项目的工作旨在开发评估跨越物理尺度的结构的稳定性的技术。在不同的尺度上可以获得各种信息，目标是开发将这些不同的信息拼接在一起以回答相关的物理问题的方法。先进技术和通信中的问题通常会利用材料的光学特性。这项提案中描述的工作将为现有和承诺的技术的关键结构提供技术，如高比特率通信和玻色-爱因斯坦凝聚体。在更一般的层面上，即将开发的技术有望为高维结构和图案的稳定性确定提供一个新的观点。另一个方面是培养跨学科数学的研究生和博士后。这项工作将进一步推动这一努力，并使培训学生从事跨学科工作的一些关键活动系统化。特别是，在与领域科学家合作的团队中，重点将致力于教授问题识别能力。代表不足的少数群体的参与也将是一个目标，预计这一努力将使一些少数群体和妇女在现代跨学科数学中达到具有竞争力的水平。

Proposal: DMS-0410267PI: Christopher K. JonesInstitution: University of North Carolina Chapel HillTitle: Permanent Structures across Scales: Assessing Stability and the Effects of PerturbationsABSTRACT: The analysis of structures that span scales pose challenges to mathematics that have led in recent years to new ideas and theories. The investigator in this work focuses on issues associated with the stability of permanent structures that have contrasting structure at the different scales. Their presence in a given system is the result of a balance between nonlinearity and a spatio-physical effect such as dispersion (e.g., for a solitary wave in optical communications), diffusion (e.g., propagating waves in biological media) or convection (e.g., shocks in materials or gases). The issue of stability requires a deep analysis that can reveal more intricate features in the inner workings of a given system. The work of this project will address four issues: (1) the determination of the stability for waves that are held together on different scales by different spatio-physical effects; (2) The assessment of the stability of patterns in multi-dimensions that have substructures on different scales; (3) The effects of stability at the macro-scale on controlling variations on a smaller scale; and (4) For complex structures, the best way to measure the manifestations of instability that influence a range of scales.Critical questions in a physical or biological model often concern the stability of a permanent structure, such as a nonlinear wave or pattern. The work of this project aims at developing techniques for assessing the stability of structures that cross physical scales.A variety of information may be available at the different scales and the goal is to develop approaches for patching together these contrasting pieces of information to answer the relevant physical questions. Problems in advanced technology and communications routinely exploit the optical properties of materials. The work described in this proposal will supply techniques for key structures of existing and promised technologies, such as high bit-rate communications and Bose-Einstein condensates. At a more general level, the techniques to be developed promise a new viewpoint on stability determinations for higher-dimensional structures and patterns. Another dimension is the training of graduate students and postdocs in interdisciplinary mathematics. This work will both further that effort and systematize some of the key activities in the training of students for interdisciplinary work. In particular, a focused effort will be aimed at teaching problem discernment while working in a team with domain scientists. The involvement of under-represented minorities will also be a goal and it is anticipated that this effort will bring a number of minorities and women to a competitive level in modern interdisciplinary mathematics.