Abstract: We will discuss the insulated conductivity problem with closely spaced insulators embedded in a homogeneous matrix where the current-electric field relation is the power law $J = |E|^E$ . The electric field may blow up as the distance between insulators approaches zero. We will quantitatively analyze the concentration of the electric field $E$ between the inclusions. We will also talk about the corresponding perfect conductivity problem if time permits. Joint work with Hongjie Dong (Brown University) and Zhuolun Yang (Brown University).