Describing Blaschke products by their critical points

In this talk, I will discuss a problem which originates in complex analysis but is really a problem in non-linear elliptic PDE. It is well known that up to post-composition with a Mobius transformation, a finite Blaschke product may be uniquely described by the set of its critical points. I will discuss an infinite-degree version of this problem posed by Dyakonov. Let J be the set of inner functions whose derivative lies in Nevanlinna class. I will explain that an inner function in J is uniquely determined by Inn(F’), that is, by the collection of its critical points and the singular inner factor of the derivative. In this setting, one can prescribe the critical set to be any Blaschke sequence, while the singular measure on the unit circle must satisfy a certain restriction: its support must be contained in a countable union of Beurling-Carleson sets.