Differential complexes encode important algebraic and differential structures of physics models. Different problems involve different differential structures and complexes. For grad-div-curl related problems, such as those from electromagnetism and fluid dynamics,the de Rham complex plays a fundamental role. For other problems, such as those fromcontinuum mechanics, differential geometry, and general relativity, other complexes arerequired, such as the so-called elasticity (Kröner, Calabi) complex. These complexes andtheir properties can be systematically derived from the de Rham complex via a Bernstein–Gelfand–Gelfand (BGG) construction. There appears to be a neat correspondence between a large class of continuum mechanics models and the BGG machinery. Hence, differential complexes also provide a new angle for developing mechanics models and shed light on their structure-aware formulation. In this talk, we discuss the BGG machinery and their correspondence to elasticity, microstructures (micropolar models), continuum defects, dimension reduction, and multi-dimensional models. This paves a way for structure-preserving discretization.

      胡凯博，北京大学数学博士（2017年），曾在奥斯陆大学和明尼苏达大学从事博士后研究，并曾任牛津大学胡克研究员。现为爱丁堡大学皇家学会大学研究员。他荣获2023年SIAM计算科学与工程早期职业奖（SIAM Early Career Award）及2024年欧盟ERC起步基金（ERC Starting Grant）。他与张倩和张智民合著的论文被纳入SIAM高影响力文章集；与Douglas Arnold合著的论文《Complexes from complexes》获得2025年ICBS科学前沿奖（ICBS Frontier of Science Award）。他的研究兴趣包括保持结构的有限元方法、离散几何与拓扑及其应用。