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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\mathcal{O}_{n+1} = \\C\\{x, y, z_2, \\dots, z_n\\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]] in $n + 1$ variables.\nTwo germs $f, g \\in \\mathcal{O}_{n+1}$ are *right equivalent* if $g = f \\circ \\phi$ for a biholomorphic germ $\\phi$ of $(\\C^{n+1}, 0)$.\nLet $f$ be a germ with $f(0) = 0$ and an [[isolated singularity|VTGYEES3VS]] at $0$.\nIt is *simple* if, for $k$ large, a neighbourhood of its $k$-jet in the space of $k$-jets meets only finitely many right equivalence classes.\nWrite $Q = z_2^2 + \\cdots + z_n^2$.\nThen $f$ is simple if and only if it is right equivalent to exactly one germ of the following list:\n\n1. $A_k$: $x^{k+1} + y^2 + Q$, for $k \\ge 1$;\n2. $D_k$: $x^2 y + y^{k-1} + Q$, for $k \\ge 4$;\n3. $E_6$: $x^3 + y^4 + Q$;\n4. $E_7$: $x^3 + x y^3 + Q$;\n5. $E_8$: $x^3 + y^5 + Q$.\n\n## Remarks\n\nVladimir Arnold proved the classification in 1972.\nIn each case the index is the [[Milnor number|1ZE51P7GRQ]]: $\\mu(A_k) = \\mu(D_k) = k$ and $\\mu(E_k) = k$.\nThe proof uses finite determinacy to reduce to polynomials and the splitting lemma to remove the nondegenerate quadratic part $Q$.\nThen a case analysis of the low-degree terms of the remaining function of one or two variables finishes it.\nA function of two variables whose $3$-jet is zero is never simple: it lies near the family $x^4 + t x^2 y^2 + y^4$, whose cross-ratio of the four lines is a continuous modulus.\n\nExample: $f = x^3 + y^3$ has $\\mu = 4$ and is right equivalent to $D_4$, since its cubic part has three distinct linear factors, as does $x^2 y + y^3 = y(x^2 + y^2)$.\n\n## Explanations\n\n### L1 human\n\nMost critical points of functions are simple, like the bottom of a bowl.\nArnold asked which singular functions cannot be continuously deformed into infinitely many inequivalent nearby types.\nThe answer is a short list, labelled by letters $A$, $D$, $E$ and an index.\nThe same letters label the simply-laced Dynkin diagrams of Lie theory, and this is no accident.\n\n### L2 human\n\nThe simple singularities appear in many guises.\nFor $n = 2$, they are the [[ADE singularities|HQNXQBZ6C7]], the surface singularities $\\C^2 / G$ for finite subgroups $G$ of $\\mathrm{SL}_2(\\C)$, also called Kleinian or Du Val singularities.\nFor $n = 2$, the intersection form on $H_2$ of the Milnor fibre is, up to sign, the Cartan matrix of the Dynkin diagram of the same name.\nArnold continued with the unimodal and bimodal singularities, which include the $14$ exceptional unimodal singularities.\nThe elementary catastrophes of [[catastrophe theory|6FKJ6SXP3G]] are real forms of simple singularities of small Milnor number.\n",
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