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My question is whether the archetype of 'wild' problems in algebra, namely classifying pairs of square matrices up to similarity, is 'non-smooth' in the sense of Borel reducibility. This was ...

$\DeclareMathOperator\svd{svd}$Consider the SVD of rectangular matrices as operators $$\svd:\mathbb C^{n\times m}\to \mathbb U_n\times \mathbb D_{n, m}\times\mathbb U_m$$ where $\mathbb U_n$ is the ...