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  "body": "\n## Summary\n\nSingularity theory studies the points where a variety or a map fails to be smooth, and classifies them up to local changes of coordinates.\nIts objects are germs: the behaviour of a function, a map or a variety near one point.\nMain tools are invariants such as the [[Milnor number|1ZE51P7GRQ]], and resolutions by [[blowing up|C4KGYAFPN7]].\nIt connects algebraic geometry with differential topology, through [[Morse functions|KPQ96640ZJ]] and catastrophe theory.\n\n## Explanations\n\n### L1 human\n\nUnder a microscope, most points of a curve or surface look like a flat piece of space.\nA few may not, such as the tip of a cone or the crossing point of a figure eight.\nSingularity theory asks what such special points can look like, and when two of them are the same up to a smooth change of coordinates.\nFor functions, the singular points are the [[critical points|0YTCMFB4HC]].\nOver 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.\nDegenerate ones, such as $x^3$ or $x^2 + y^3$, give a rich classification.\n",
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