The homological permanent–determinant conjecture

Let $h(\mathrm{perm}_n)$ and $h(\mathrm{det}_n)$ denote the homological complexities associated with the $n\times n$ permanent and determinant, respectively, and let $\mathcal{VP}$ be the class of polynomial-size algebraic circuits. Homological permanent–determinant conjecture. The permanent requires super-polynomial algebraic circuits if and only if

$$\lim_{n\to\infty}\frac{h(\mathrm{perm}_n)}{h(\mathrm{det}_n)}=\infty.$$

Moreover, $\mathrm{perm}\notin\mathcal{VP}$ is equivalent to $h(\mathrm{perm}_n)$ growing super-polynomially in $n$. This proposes a homological reformulation of the permanent-versus-determinant problem; the source gives no resolution of either equivalence.