Kaibo Hu (Mathematical Institute, University of Oxford) – Complexes from complexes: continuous and discrete

The Bernstein–Gelfand–Gelfand (BGG) sequences provide a framework for encoding the differential structure of tensor fields with symmetries, generalizing the de Rham complex to higher-order and symmetric tensors. These complexes underlie important examples such as the Calabi complex in differential geometry and the Kröner complex in continuum mechanics, yielding cohomological information and decomposition properties essential for well-posedness in related PDE systems.

We present a systematic procedure to derive new differential complexes from existing ones, including BGG-type sequences, and establish analytic properties. Building on this, we focus on form-valued forms (double forms), which include fields like the metric tensor and curvature tensor. We discuss finite element discretizations of these structures, extending Whitney forms and compatible complexes to canonical discretizations of general symmetric tensors, enabling discrete differential-geometric structures and tensor decompositions in 2D, 3D, and higher dimensions with applications in computational electromagnetics, elasticity, and linearized gravity.