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| A picture of the complete graph K_n consists of n labelled points in the
plane, connected with (n choose 2) lines. I'm going to talk about the
algebraic relations that must hold among the slopes of these lines.
This sounds like a problem in classical geometry, but it turns out that
the tools to attack it come from combinatorics. First, the equations
defining a picture can be described using the theory of combinatorial
rigidity of graphs. Second, once one knows what these equations are, one
can apply another combinatorial idea, the theory of Stanley-Reisner rings,
to obtain geometric invariants of the space of all solutions. Finally,
various sorts of labelled trees play important roles in describing these
invariants combinatorially.
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