Geometry and Combinatorics of Rigidity Theory and its Applications

刚度理论的几何与组合学及其应用

• 批准号: RGPIN-2015-04624
• 负责人: Whiteley, Walter
• 金额: $ 1.24万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613073
• 关键词: Geometry; Combinatorics; Rigidity; Theory; its

We live in 3D and we design and build in 3D. The constraints on the structure of the successful designs, and the analysis of the failures of designs, involve 3D geometry in its various forms, as well as its simplified form as combinatorics: counting the parts, counting the numbers and patterns of connections among the parts. Mathematicians find novel ways to assemble components, and to analyze the way nature has assembled the parts. Such structures then raise a number of questions that we work on. Across many fields of Engineering and Science the same core questions arise about the range of realizations of structures satisfying a set of conditions or constraints. Sometimes there are no structures with these values. Sometimes there is just one (global uniqueness). Sometimes there are several realizations, but they are locally unique (rigid). Sometimes there is a continuous path of realizations (the structure is flexible). The methods being refined in this project address all of these variations of rigidity, both in the plane and in 3-space. The ultimate goal is have a computer algorithm that is able to test a given set of values and structures for any of these properties, in a reasonable time. We have algorithms for some of these, some of which work almost all of the time – but there is room for failures of the ‘general’ algorithm due to special geometry of the patterns in a specific set of values. For example, the structure may have symmetry – because multiple identical copies of a substructure are being combined, either in built structures or in biological structures such as proteins and viruses. Identifying these ‘special positions’ is currently a focus of the research, as is working out the impact of those special positions on the positive functioning of biological structures and machines or buildings as well as the potential for failures both in built structures (such as buildings and bridges) and in biological interventions, such as drug design.  Core to this work are international collaborations which bring in multiple areas of mathematics, such as modern combinatorial geometry (matroid theory) and older areas such as projective geometry over multiple spaces (Euclidean, spherical) and into other more exotic areas (such as hyperbolic geometry) to refine our understanding of the basic structures and the structure of the constraint systems. The project combines mathematical results with ongoing work with innovative practitioners in several field of engineering (mechanical and civil), computer science, including robotics, computer-aided design (CAD) and use of sensor networks, and several fields of science (material science and biochemistry). All of these interactions have the potential to inform professional practices, and improve the predictions of the behavior of the systems.

我们生活在 3D 环境中，我们在 3D 环境中进行设计和构建。成功设计的结构约束以及设计失败的分析涉及各种形式的 3D 几何及其简化形式的组合：计算数学家们找到了组装部件的新方法，并分析了大自然组装部件的方式，然后提出了我们正在研究的许多问题。 在工程和科学的许多领域中，都会出现关于满足一组条件或约束的结构的实现范围的问题，有时没有具有这些值的结构，有时只有一个（全局唯一性）。实现，但它们是局部唯一的（刚性的），有时存在连续的实现路径（结构是灵活的）。 该项目中正在完善的方法解决了平面和三维空间中的所有这些刚性变化，最终目标是拥有一种能够测试任何给定值和结构的计算机算法。这些属性，在合理的时间内，我们有针对其中一些的算法，其中一些几乎在所有时间都有效 - 但由于特定组中图案的特殊几何形状，“通用”算法有失败的空间。例如，结构可能具有对称性——因为有多个相同的副本。子结构的各个部分正在被组合起来，无论是在建筑结构中还是在蛋白质和病毒等生物结构中，识别这些“特殊位置”是目前研究的重点，同时研究这些特殊位置对子结构积极功能的影响。生物结构和机器或建筑物，以及建筑结构（例如建筑物和桥梁）和生物干预（例如药物设计）中发生故障的可能性。 这项工作的核心是国际合作，将数学的多个领域（例如现代组合几何（拟阵理论））和较旧的领域（例如多个空间上的射影几何（欧几里得、球面））以及其他更奇特的领域（例如双曲几何）引入进来。 ）以完善我们对基本结构和约束系统结构的理解。 该项目将数学结果与多个工程领域（机械和土木）、计算机科学（包括机器人技术、计算机辅助设计（CAD）和传感器网络的使用）以及多个科学领域（材料科学和所有这些相互作用都有可能为专业实践提供信息，并改善对系统行为的预测。