Geometry and Combinatorics of Rigidity Theory and its Applications

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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.