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Terry Tao
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In higher order Fourier analysis, there are d-dimensional parallelopiped structures which can be viewed as a $2^d-1$-ary relation of the form "given all but one vertex of a parallelopiped as input, return the final vertex as output". (Here "parallelopiped" should be interpreted in a suitably abstract sense, as an operation obeying a certain number of axioms.) This point of view is taken for instance in this paper of Camarena and Szegedy, building on the earlier work of Host and Kra. In the d=2 case, this ternary operation is essentially equivalent to an additive operation once one fixes an origin (in which case the ternary operation becomes $(x,y,z) \mapsto x-y+z$). For d=3, these operations are governed by 2-step nilpotent groups, and more generally d-dimensional parallelopiped structures are governed by d-1-step nilpotent groups.

These parallelopiped structures share some formal resemblances to cubic complexes, which are constructs that appear mostly in algebraic topology and are discussed for instance here.

Terry Tao
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  • 539