Both Morse theory and Lusternik-Schnirelmann theory link algebra, topology and analysis in a geometric setting. The two theories can be formulated in finite geometries like graph theory or within finite abstract simplicial complexes. We work here mostly in graph theory and review the Morse inequalities b(k)-b(k-1) + ... + b(0) less of equal than c(k)-c(k-1) + ... + c(0) for the Betti numbers b(k) and the minimal number c(k) of Morse critical points of index k and the Lusternik-Schnirelmann inequalities cup+1 less or equal than cat less or equal than cri, between the algebraic cup length cup, the topological category cat and the analytic number cri counting the minimal number of critical points of a function.

翻译：莫尔斯理论与柳斯捷尔尼克-施尼雷尔曼理论均在几何框架下将代数、拓扑与分析联系起来。这两种理论可以在有限几何（如图论）或有限抽象单纯复形中表述。本文主要在图论框架下展开，回顾了关于贝蒂数 b(k) 与指数 k 的莫尔斯临界点最小数目 c(k) 的莫尔斯不等式 b(k)-b(k-1) + ... + b(0) ≤ c(k)-c(k-1) + ... + c(0)，以及联系代数杯积长度 cup、拓扑畴数 cat 与记录函数临界点最小数目的解析量 cri 的柳斯捷尔尼克-施尼雷尔曼不等式 cup+1 ≤ cat ≤ cri。