Bernd Schulze

Structural rigidity analysis via group theory and homology

Structural rigidity examines the rigidity and flexibility of bar-joint frameworks and related geometric constraint systems, blending combinatorial and geometric perspectives. Its origins trace back to classical works of Euler, Cauchy and Maxwell on the rigidity of polyhedra and skeletal frames. Within this field, graphic statics - a geometry-based subfield - serves as a powerful toolbox for structural analysis and design. Both areas have experienced a resurgence of interest, fuelled by diverse modern applications in engineering, robotics, CAD, structural biology, and materials science.

In this talk, I will introduce structural rigidity and graphic statics, focusing on key combinatorial results and problems for bar-joint frameworks, as well as the classical Maxwell-Cremona correspondence from graphic statics linking self-stressed frameworks to dual force diagrams and polyhedral liftings. I will then show how basic group representation theory can be used to obtain significant additional information about infinitesimal motions and self-stresses of symmetric frameworks. Recently, in collaboration with the company SOM, these methods have been applied to create new design tools for material-efficient long-span structures such as gridhell roofs. Finally, I will show that the newly developed theory of cosheaf homology can be used to refine the Maxwell-Cremona correspondence by incorporating symmetry.