FRG: Collaborative Research: Stability of Structures Large and Small

FRG：合作研究：大大小小的结构的稳定性

• 批准号: 1564480
• 负责人: Meera Sitharam
• 金额: $ 29.92万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-15 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1564480&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Stability; Structures

This award supports collaborative research efforts in the area of materials science by an interdisciplinary team comprising pure mathematicians, applied mathematicians, computer scientists, and physicists. Answers to natural questions about the stability and rigidity of material structures involve understanding the geometry of their components. Recent advances in materials synthesis have emphasized the need for a deeper understanding of the geometric stability of physical structures at the atomic scale and the need for insight at all scales, from atomic to macroscopic. Key mathematical tools for this analysis come from the area of "rigidity theory," which studies the mathematical properties of discrete sets of points with the distances between certain pairs of points held fixed or constrained by distance inequalities. Rigidity theory lies at the nexus of discrete geometry, graph theory, and algorithms, and it has deep connections to semidefinite programming and convex geometry. This project aims to deepen understanding of the stability of material structures. One goal of this project is to develop a mechanistic explanation of tunneling between asymmetric stable configurations of two-dimensional disordered materials, such as glass. Construction of accurate mechanistic and computational models requires deep mathematical analysis and development of appropriate algorithms. A second goal is to develop methods for predicting the stability, configurational entropy, and kinetics of small short-ranged-potential systems in three dimensions. Examples of such distance-constraint systems include small molecular structures, as well as colloidal clusters, containing a few particles bound together by reversible attractive interactions, modeled as sticky spheres. What kinds of rigid configurations are there, and what are computationally feasible tests for their rigidity? How do these particles move and the structures deform? There is a tight link between rigidity theory and the general convexity and duality properties of the positive semidefinite cone, a central concept in numerical optimization. Finding a recursive decomposition of a generically rigid framework into rigid subsystems is a longstanding problem. Additionally, matroid theory, important in rigidity theory, has made the characterization of rigid systems more approachable and more algorithmically efficient. Rigidity theory could have implications for algorithms for low-rank matrix completion as well. These connections, questions, and implications will be explored in this project.

该奖项支持由纯数学家，应用数学家，计算机科学家和物理学家组成的跨学科团队在材料科学领域的合作研究工作。关于材料结构的稳定性和刚性的自然问题的答案涉及了解其组件的几何形状。 材料合成的最新进展强调，需要更深入地了解原子量表的物理结构的几何稳定性，以及从原子能到宏观的各种规模的洞察力的需求。该分析的关键数学工具来自“刚度理论”的领域，该区域研究了离散点集的数学特性，这些点集的某些点固定或受距离不等式约束的点之间的距离之间的距离。刚性理论在于离散几何学，图理论和算法的联系，它与半芬矿编程和凸几何形状有着深厚的联系。 该项目旨在加深对材料结构的稳定性的理解。该项目的一个目标是开发一种机械解释，对二维无序材料（例如玻璃）的不对称稳定构型（例如玻璃）之间的隧穿。精确的机械和计算模型的构建需要深入的数学分析和适当算法的开发。第二个目标是开发用于预测三个维度小型短效率系统的稳定性，配置熵和动力学的方法。此类距离构成系统的示例包括小分子结构以及胶体簇，其中包含一些粒子，由可逆的有吸引力的相互作用结合在一起，以粘性球体建模。有哪种刚性配置，对于它们的刚性，哪些计算可行测试是什么？这些颗粒如何移动和结构变形？刚性理论与阳性半圆锥锥的一般凸度和二元性能之间存在紧密联系，这是数值优化的中心概念。将一般刚性框架递归分解为刚性子系统是一个长期存在的问题。此外，在刚性理论中重要的矩形理论使刚性系统的表征更加平易近，算法更有效。刚性理论也可能对算法对低级别矩阵完成也有影响。这些项目将探讨这些联系，问题和含义。