Geometry and Combinatorics of Rigidity Theory and its Applications

我们生活在3D中，我们在3D中设计和建造。 对成功设计的结构的约束，以及对设计失败的分析，涉及各种形式的3D几何，以及其简化形式的组合学：计算部件，计算部件之间的连接的数量和模式。 数学家找到了新的方法来组装组件，并分析自然界组装部件的方式。 这样的结构然后提出了一些问题，我们的工作。* 在工程和科学的许多领域，同样的核心问题出现了关于满足一组条件或约束的结构的实现范围。 有时候没有具有这些值的结构。 有时只有一个（全局唯一性）。 有时有几个实现，但它们是局部唯一的（刚性的）。 有时有一个连续的实现路径（结构是灵活的）。 * 本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结构的任何这些属性。 我们有算法，其中一些工作几乎所有的时间-但有空间失败的“一般”算法，由于特殊的几何图案在一组特定的值。 例如，结构可能具有对称性-因为子结构的多个相同副本正在组合，无论是在构建的结构中还是在生物结构（如蛋白质和病毒）中。 确定这些“特殊位置”是目前研究的一个重点，研究这些特殊位置对生物结构和机器或建筑物的积极运作的影响，以及在建筑结构（如建筑物和桥梁）和生物干预（如药物设计）中失败的可能性。 * 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（拟阵理论）和旧的领域，如多个空间上的射影几何（欧几里德，球面）和其他更奇异的领域（如双曲几何），以完善我们对基本结构和约束系统结构的理解。 * 该项目将数学成果与工程（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等多个领域的创新实践者正在进行的工作相结合，以及多个科学领域（材料科学和生物化学）。所有这些交互都有可能为专业实践提供信息，并改善对系统行为的预测。

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.