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，涉及到各种形式的三维几何，以及其作为组合学的简化形式：计算部件，计算部件之间的连接数量和模式。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，这样的结构提出了一些我们需要解决的问题。*在工程和科学的许多领域，同样的核心问题出现了，即满足一组条件或约束的结构的实现范围。有时，不存在具有这些价值观的结构。有时只有一个(全球唯一性)。有时有几种实现，但它们是局部唯一的(严格的)。有时会有一条连续的实现路径(结构是灵活的)。*本项目中改进的方法解决了所有这些在平面和三维空间中的刚性变化。最终目标是有一种计算机算法，能够在合理的时间内测试这些属性中的任何一组给定值和结构。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--但由于模式在特定的一组值中的特殊几何形状，“通用”算法有失败的空间。例如，结构可能具有对称性--因为一个亚结构的多个相同副本被组合在一起，无论是在构建的结构中，还是在蛋白质和病毒等生物结构中。这项工作的核心是引入多个数学领域的国际合作，如现代组合几何(拟阵理论)和多空间上的射影几何(欧几里得，并深入到其他更奇特的领域(如双曲几何)，以加深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几个工程领域(机械和土木工程)、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)的创新从业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.