Geometry and Combinatorics of Rigidity Theory and its Applications

我们生活在3D中，我们用3D设计和建造。成功设计的结构约束和设计失败的分析，涉及到各种形式的三维几何，以及其简化形式的组合学：计算零件，计算零件之间连接的数量和模式。数学家们发现了组装部件的新方法，并分析了大自然组装部件的方式。这样的结构提出了一些我们正在研究的问题。在工程和科学的许多领域中，同样的核心问题是关于满足一系列条件或约束的结构的实现范围。有时不存在具有这些值的结构。有时只有一个（全局唯一性）。有时有几种实现，但它们在局部是唯一的（刚性的）。有时有一个连续的实现路径（结构是灵活的）。***在这个项目中改进的方法解决了平面和三维空间中所有这些刚度的变化。最终目标是有一个计算机算法，能够在合理的时间内测试给定的一组值和任何这些属性的结构。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-但由于特定值集合中模式的特殊几何形状，“通用”算法有失败的余地。例如，结构可能具有对称性——因为在建筑结构或蛋白质和病毒等生物结构中，一个亚结构的多个相同副本正在组合。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生物结构、机器或建筑物的积极功能的影响，以及在建筑结构（如建筑物和桥梁）和生物干预（如药物设计）中潜在的失败。***这项工作的核心是国际合作，它带来了数学的多个领域，如现代组合几何（矩阵理论）和旧的领域，如投影几何在多个空间（欧几里德，球面）和其他更奇特的领域（如双曲几何），以完善我们对基本结构的理解和约束系统的结构。***该项目将数学结果与几个工程领域（机械和土木）、计算机科学（包括机器人、计算机辅助设计（CAD）和传感器网络的使用）以及几个科学领域（材料科学和生物化学）的创新实践者正在进行的工作结合起来。所有这些交互都有可能为专业实践提供信息，并改进对系统行为的预测。

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.