Valiant's VP versus VNP conjecture

Let $\mathfrak S_m$ denote the permutation group on $m$ elements, let $y^i_j$ be linear coordinates on $\mathbb C^{m^2}$, and define the permanent by

$$\operatorname{perm}_m(y^i_j)=\sum_{\sigma\in\mathfrak S_m} y^1_{\sigma(1)}\cdots y^m_{\sigma(m)}.$$

A polynomial-size circuit is a circuit whose size is bounded by a polynomial in the input degree. Valiant's conjecture. There does not exist a polynomial-size circuit computing the permanent. This is the central separation conjecture of algebraic complexity theory and motivates comparing the permanent with the determinant; its status is open.