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  "body": "\n## Summary\n\nThe Thom–Boardman singularities sort the [[singular points|1JG3HFWX3T]] of a smooth map by a sequence of numbers $(i_1, i_2, \\ldots)$.\nThe first number is the dimension of the kernel of the differential.\nThe second is the same dimension for the restriction of the map to the set of points with that first number, and so on.\nThom introduced the first steps, and Boardman made the construction work in general by defining the corresponding subsets of jet space.\nFor a generic map the sets $\\Sigma^{i_1, i_2, \\ldots}(f)$ are submanifolds of known codimension, and they give a first coarse classification of singularities.\n\n## Explanations\n\n### L1 human\n\nFor a typical map between manifolds, first look at how many directions the map crushes at each point; that number is $i_1$.\nThe points where exactly $i_1$ directions are crushed form a curve, a surface, or some other smaller manifold.\nNow restrict the map to this smaller manifold and ask again how many directions it crushes there; that number is $i_2$.\nFor a map of the plane to the plane, a fold point has symbol $(1, 0)$: one direction is crushed, but the map is regular along the fold curve.\nA cusp point has symbol $(1, 1, 0)$: the map also crushes the direction along the fold curve at that point.\n",
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