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  "body": "\n## Summary\n\nCritical point theory studies the [[critical points|0YTCMFB4HC]] of a differentiable function on a manifold.\nIts tool is the topology of the manifold and of the sublevel sets of the function.\nIts main branches are Morse theory, which uses [[nondegenerate critical points|MGPPCDFQDG]] and their [[indices|MAX2ZYG6H9]], and [[Lusternik–Schnirelmann theory|QARH15D2PN]], which uses minimax values.\nIn infinite dimensions it needs a compactness hypothesis such as the [[Palais–Smale condition|4724KRNHAT]].\nIt then gives existence results for geodesics, harmonic maps and nonlinear elliptic equations.\n\n## Explanations\n\n### L1 human\n\nA critical point of a function is a point where its derivative vanishes, such as a peak, a pit or a saddle of a landscape.\nCritical point theory turns topology into existence theorems: if the shape of the space is complicated, the function must have many critical points.\nFor example, any smooth function on a circle has at least two critical points, a maximum and a minimum.\nOn a torus it has at least three, and a Morse function on a torus has at least four.\n",
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