Many facets of cohomology: Differential complexes and structure-aware formulations
Kaibo Hu
复杂性和同质学,传统上是拓扑学的核心,已成为应用数学和科学的基本工具。 本调查探讨了他们在不同领域的角色,从偏微分方程和连续力学到爱因斯坦方程和网络理论的重新计算。 受兼容和结构保护离散化(如有限元素外部微积分(FEEC))的进步激励,我们研究了微分复合物如何编码关键特性,如微分方程解决方案的存在,唯一性,稳定性和刚性。 我们证明,固体和流体力学中的各种基本概念和模型本质上是在微分复合物方面制定的。
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 proper...