Many facets of cohomology: Differential complexes and structure-aware formulations
Kaibo Hu
TL;DR
The paper articulates a unifying framework where differential complexes organize the relationships among physical observables across disciplines. By detailing De Rham and BGG complexes, twisted and conformal variants, and their analytic consequences via Hodge theory, it shows how exactness and finite-dimensional cohomology govern existence, stability, and rigidity of solutions. It then connects these ideas to solid mechanics, fluids, general relativity, and graphs, illustrating how structure-aware formulations yield robust, physics-faithful models and discretizations. The work emphasizes that preserving cohomology in discretizations (e.g., FEEC and related BGG constructions) is key to well-posed, stable computations, and it highlights cross-domain insights such as elasticity from Cosserat models, MHD helicity preservation, and discrete topological data analysis. Overall, the differential-complex perspective provides a cohesive toolkit for modelling, analysis, and numerics in a wide range of scientific contexts.
Abstract
Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum mechanics to reformulations of the Einstein equations and network theory. Motivated by advances in compatible and structure-preserving discretisation such as Finite Element Exterior Calculus (FEEC), we examine how differential complexes encode critical properties such as existence, uniqueness, stability and rigidity of solutions to differential equations. We demonstrate that various fundamental concepts and models in solid and fluid mechanics are essentially formulated in terms of differential complexes.