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Many facets of cohomology: differential complexes and structure-aware formulation
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DOI：10.1098/rsos.250728.png)
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Abstract
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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 PDEs and continuum mechanics to reformulations of the Einstein equations and network theory. Motivated by advances in compatible and structure-preserving discretization 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.
Keywords:
differential complex
cohomology
elasticity
de Rham complex
Bernstein-Gelfand-Gelfand construction
helicity
microstructure
numerical relativity
topological data analysis
Hodge Laplacian
Journal
IF:
2.9
Papers:
775
Citations:
1.8W
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