Two problems on coinvariant subspaces of the shift operator

Two problems are posed that involve the star-invariant subspace $K_{\theta }^p$ (in the Hardy space $H^p$) associated with an inner function $\theta$. One of these asks for a characterization of the extreme points of the unit ball in $K_{\theta }^{\mathrm{\infty }}$, while the other concerns the Fermat equation $f^n+g^n=h^n$ in $K_{\theta }^p$.