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  "body": "\n## Summary\n\nThe ADE singularities, also called Du Val singularities, simple surface singularities or rational double points, are the simplest singular points of complex surfaces.\nThey are the quotients $\\C^2/G$ by finite subgroups $G$ of the [[special linear group|0V9PMN0RBH]] $SL(2, \\C)$.\nThey come in two infinite families $A_n$ and $D_n$ and three exceptional cases $E_6$, $E_7$, $E_8$, matching the simply laced Dynkin diagrams.\nThe match is visible in the minimal resolution, whose exceptional curves meet in the pattern of the Dynkin diagram.\n\n## Explanations\n\n### L1 human\n\nThe simplest example is the cone $x^2 + y^2 + z^2 = 0$, the $A_1$ singularity, which is the quotient of the plane by $v \\mapsto -v$.\nBlowing up the tip of the cone replaces it by one smooth rational curve.\nIn general the $A_n$ singularity is resolved by a chain of $n$ such curves, each meeting the next one, just like the Dynkin diagram $A_n$.\nThe same ADE list classifies finite rotation groups in three dimensions and simple Lie algebras of simply laced type.\nThis is why these singularities are a crossroads of mathematics.\n",
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