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  "body": "\n## Statement\n\nLet $m \\ge 2$ and $n \\ge 1$ be integers.\nLet $M$ be a smooth [[Riemannian manifold|7CBT4SM83Y]] of dimension $m + n$, and let $T$ be an $m$-dimensional [[integral current|EG4AJVEFT2]] in $M$ that is [[area-minimizing|4NKP73KVVW]].\nLet $\\operatorname{sing} T$ be the set of points of the support of $T$, outside the support of its boundary, with this property.\nNear the point, the support of $T$ is not a smooth embedded submanifold.\nThen the [[Hausdorff dimension|SSNFTHZ6YJ]] of $\\operatorname{sing} T$ is at most $m - 2$.\n\n## Remarks\n\nFrederick Almgren proved the theorem in a very long work that was published in 2000, after his death.\nCamillo De Lellis and Emanuele Spadaro later gave a shorter proof.\nThe proof studies blow-ups at points where the tangent cone is a plane with integer multiplicity $Q \\ge 2$.\nNear such points the current is approximated by the graph of a $Q$-valued function that minimizes a Dirichlet energy.\nA frequency function controls the rate at which the current departs from the plane.\nThe bound $m - 2$ is sharp in higher codimension.\nBy Wirtinger's inequality, complex algebraic curves in $\\C^2 = \\R^4$ are area-minimizing, and the curve $\\set{(z, w) : z^2 = w^3}$ has a singular point at the origin, a set of dimension $0 = 2 - 2$.\nIn codimension $n = 1$ the singular set is much smaller: its dimension is at most $m - 7$, by the work of Federer and Simons.\n\n## Explanations\n\n### L1 human\n\nAn area-minimizing surface of any dimension, in a space of any dimension, can have singular points where it is not smooth.\nAlmgren's theorem says that these singular points form a set that is small: two dimensions smaller than the surface itself.\nFor two-dimensional surfaces, the singular points are isolated, as for complex curves crossing themselves.\n\n### L2 human\n\nIn codimension one, regularity follows from De Giorgi's excess-decay method, because the tangent planes have multiplicity one.\nIn higher codimension a tangent plane can carry multiplicity, so branching phenomena occur, and De Giorgi's method no longer applies.\nAlmgren's theorem handles this with $Q$-valued functions and the frequency function, tools that have since spread to free boundary problems.\n",
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