Index theory for subfactors. Given an inclusion of subfactors $N \subseteq M$, there is an index $[M:N]$, which a priori is just a positive real number. Vaughan Jones showed that the index is constrained in the values it can take: it can be any real number $\geq 4$, or it can be of the form $4\cos^2(\pi/n)$ for some $n \ge 3$.
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