---
tag: 42V5XGZS4F
kind: concept
name: Singularity theory
msc: [14B05, 32S05, 58K05]
sources:
  - url: https://en.wikipedia.org/wiki/Singularity_theory
same_as:
  wikidata: Q4245771
  wikipedia: Singularity theory
---

## Summary

Singularity theory studies the points where a variety or a map fails to be smooth, and classifies them up to local changes of coordinates.
Its objects are germs: the behaviour of a function, a map or a variety near one point.
Main tools are invariants such as the [[Milnor number|1ZE51P7GRQ]], and resolutions by [[blowing up|C4KGYAFPN7]].
It connects algebraic geometry with differential topology, through [[Morse functions|KPQ96640ZJ]] and catastrophe theory.

## Explanations

### L1 human

Under a microscope, most points of a curve or surface look like a flat piece of space.
A few may not, such as the tip of a cone or the crossing point of a figure eight.
Singularity theory asks what such special points can look like, and when two of them are the same up to a smooth change of coordinates.
For functions, the singular points are the [[critical points|0YTCMFB4HC]].
Over the complex numbers, near a nondegenerate critical point, a function is $x_1^2 + \cdots + x_n^2$ after a change of coordinates, so all of them look alike.
Degenerate ones, such as $x^3$ or $x^2 + y^3$, give a rich classification.