---
tag: 9ZQC7ARWTT
kind: concept
name: Critical point theory
msc: ["58E05", 47J30, 57R70]
sources:
  - url: https://en.wikipedia.org/wiki/Critical_point_(mathematics)
---

## Summary

Critical point theory studies the [[critical points|0YTCMFB4HC]] of a differentiable function on a manifold.
Its tool is the topology of the manifold and of the sublevel sets of the function.
Its main branches are Morse theory, which uses [[nondegenerate critical points|MGPPCDFQDG]] and their [[indices|MAX2ZYG6H9]], and [[Lusternik–Schnirelmann theory|QARH15D2PN]], which uses minimax values.
In infinite dimensions it needs a compactness hypothesis such as the [[Palais–Smale condition|4724KRNHAT]].
It then gives existence results for geodesics, harmonic maps and nonlinear elliptic equations.

## Explanations

### L1 human

A critical point of a function is a point where its derivative vanishes, such as a peak, a pit or a saddle of a landscape.
Critical point theory turns topology into existence theorems: if the shape of the space is complicated, the function must have many critical points.
For example, any smooth function on a circle has at least two critical points, a maximum and a minimum.
On a torus it has at least three, and a Morse function on a torus has at least four.