Flagging and colossus convergence. The rectangle (2d) was a start for the learning algorithm (in 5d). And the flag was essentially a start for the rectangle.
As another part of preparing for the digestion of the $latex E_8$ result, I need to talk about flag vareties. You’ll need at least some linear algebra to follow from this point.
A flag in a vector space is a chain of nested subspaces of specified dimensions. In three-dimensional space, for instance, one kind of flag is a choice of some plane through the origin and a line through the origin sitting inside that plane. Another kind is just a choice of a plane through the origin. The space of all flags of a given kind in a vector space can be described by solving a certain collection of polynomial equations, which makes it a “variety”. It’s sort of like a manifold, but there can be places where a variety intersects itself, or comes to a point, or has a sharp kink. In those places it doesn’t look like $latex…
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