A celebrated theorem of M.~Heins says that up to post-composition with a M\"obius transformation, a finite Blaschke product is uniquely determined by its critical points. K.~Dyakonov suggested that it may interesting to extend this result to infinite degree, however, one needs to be careful since inner functions may have identical critical sets.

Let $\mathscr J$ be the set of inner functions whose derivative lies in the Nevanlinna class. I will explain that an inner function in $\mathscr J$ is uniquely determined by the inner part of its derivative (its critical structure), and describe all possible critical structures of inner functions in $\mathscr J$. Alternatively, one may try to parameterize inner functions by 1-generated invariant subspaces of the weighted Bergman space $A^2_1$. Our technique is based on the Liouville correspondence which provides a bridge between complex analysis and non-linear elliptic PDE.