A classic problem is to find inequalities which are satisfied
by polynomials of minors of any totally positive matrix. When these
inequalities only involve multiplication, they are equivalent to finding
ratios of minors which are bounded over the set of totally positive
matrices. Furthermore, the set of matrix minors are cluster coordinates.
With M. Gekhtman and D. Soskin, we used this correspondence to classify
the bounded ratios in any finite cluster algebra. Surprisingly, the key
tool are the U-variables and "binary geometries" which have been
proposed as a tool for studying scattering amplitudes.