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.

翻译：复形与同调论，传统上是拓扑学的核心内容，现已成为应用数学与科学领域的基本工具。本综述探讨了它们在偏微分方程、连续介质力学、爱因斯坦方程的重构表述以及网络理论等多个领域中的作用。受有限元外微积分（FEEC）等相容性与结构保持离散化进展的推动，我们研究了微分复形如何编码微分方程解的存在性、唯一性、稳定性及刚性等关键性质。我们论证了固体力学与流体力学中的各种基本概念与模型本质上均以微分复形为框架进行表述。