Unique graph realisation problems under symmetry constraints

Rigidity theory is an interdisciplinary field which aims to provide techniques for understanding the rigidity and flexibility properties of discrete geometric constraint systems. The paradigmatic example is that of a bar-joint framework, which consists of stiff (fixed length) bars that are connected at their ends by freely rotational joints. Mathematically, a framework is a graph that describes how the bars are connected, paired with a mapping of the graph’s vertices (representing the joints) into d-dimensional space. The framework is "locally rigid" if any small perturbation of the joints that preserves the bar lengths results in a framework congruent to the original. However, even for a locally rigid framework, there may still be distant, non-congruent frameworks with the same bar lengths. The framework is "globally rigid" if the graph and bar lengths uniquely determine its shape in d-space, up to rigid body motions. The origins of rigidity theory date back to the work of Euler and Cauchy on the rigidity of convex polyhedra and Maxwell’s studies of articulated engineering structures. Over recent decades there has been a surge of interest in rigidity theory due to both theoretical and computational advances as well as the emergence of a range of new application areas. The geometric constraint systems studied in rigidity theory are suitable mathematical models for a wide range of both natural and engineered structures and their rigidity and flexibility is crucial for their form, behaviour and function. The local and global rigidity of a framework depends on both the underlying graph and the specific locations of the joints, and it is in general very difficult to determine whether a given framework is locally or global rigid. However, these problems become more tractable if the joints are placed generically, because in that case both global and local rigidity only depend on the combinatorics of the underlying graph. While the local rigidity theory is well-established, advances in global rigidity have only gained momentum recently, due to new mathematical breakthroughs and emerging application areas such as structural biology and sensor network localisation. Real-world structures are rarely generic and often display non-trivial symmetries, leading to unexpected rigidity or flexibility. Motivated by practical applications, a very robust theory for the local rigidity of symmetric frameworks, including a combinatorial theory for frameworks that are "symmetry-generic'' (i.e. as generic as possible with the given symmetry), has been developed in recent years. However, due to significant mathematical obstacles, the corresponding challenges in symmetric global rigidity remain largely unexplored. The aim of this project is to establish ground-breaking results regarding the global rigidity of finite symmetric and infinite periodic frameworks. The main objectives are: (1) to create algebraic-geometric certificates for symmetric frameworks to be globally rigid via the novel approach of a detailed investigation of symmetric measurement varieties, and (2) to develop a combinatorial theory for symmetry-generic global rigidity by synthesizing recently developed combinatorial tools from the non-symmetric theories of local and global rigidity. The project will yield significant theoretical results, thereby establishing a substantial new mathematical research area, while also providing tangible benefits to applied science and technology. In particular, the results will lead to new design tools for material-efficient engineering structures and improved decentralised control algorithms for autonomous multi-agent systems, and will deepen our understanding of molecular stability in biological and chemical systems.