By Hilbert's Basis Theorem, varieties in finite-dimensional spaces are cut out by a finite number of equations. Of course, this is no longer true in general for varieties in infinite-dimensional spaces. But there, some varieties with sufficiently large symmetry groups are cut out by finitely many _orbits_ of equations. This phenomenon, of which I will give several intriguing examples, ties in with beautiful topics such as algebraic statistics (where some of the examples originate), invariant theory, and the combinatorics of well-quasi-orders.
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