FRG: Collaborative Research: Stability of Structures Large and Small

该奖项支持由纯粹数学家、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。关于材料结构的稳定性和刚性的自然问题的答案涉及到对其组成部分几何结构的理解。材料合成的最新进展强调需要在原子尺度上对物理结构的几何稳定性有更深入的了解，需要从原子到宏观的所有尺度上都有洞察力。这种分析的关键数学工具来自“刚性理论”领域，它研究离散点集的数学性质，某些点对之间的距离被距离不等式固定或约束。刚性理论是离散几何、图论和算法的联系，它与半定规划和凸几何有着深刻的联系。这个项目旨在加深对材料结构稳定性的理解。这个项目的一个目标是在二维无序材料（如玻璃）的不对称稳定构型之间建立隧道的机制解释。建立精确的机械和计算模型需要深入的数学分析和开发适当的算法。第二个目标是发展在三维空间中预测小的短程势系统的稳定性、构型熵和动力学的方法。这种距离约束系统的例子包括小分子结构，以及胶体团簇，包含一些通过可逆吸引相互作用结合在一起的粒子，建模为粘性球体。有哪些刚性构型，它们的刚度有哪些计算上可行的测试？这些粒子如何移动，结构如何变形？刚性理论与正半定锥的一般凸性和对偶性之间有着密切的联系，这是数值优化中的一个核心概念。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，在刚性理论中很重要的矩阵理论使刚性系统的表征更容易接近，算法效率更高。刚性理论也可能对低秩矩阵补全算法有启示。本项目将探讨这些联系、问题和影响。

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.