On the Nevanlinna problem -- Characterization of all Schur-Agler class solutions affiliated with a given kernel

Given a domain $Omega$ in $mathbb{C}^m$, and a finite set of points $z_1,z_2,ldots, z_nin Omega$ and $w_1,w_2,ldots, w_nin mathbb{D}$ (the open unit disc in the complex plane), the textit{Pick interpolation problem} asks when there is a holomorphic function $f:Omega rightarrow overline{mathbb{D}}$ such that $f(z_i)=w_i,1leq ileq n$. Pick gave a condition on the data ${z_i, w_i:1leq ileq n}$ for such an $interpolant$ to exist if $Omega=mathbb{D}$. Nevanlinna characterized all possible functions $f$ that textit{interpolate} the data. We generalize Nevanlinnas result to a domain $Omega$ in $mathbb{C}^m$ admitting holomorphic test functions when the function $f$ comes from the Schur-Agler class and is affiliated with a certain completely positive kernel. The Schur class is a naturally associated Banach algebra of functions with a domain. The success of the theory lies in characterizing the Schur class interpolating functions for three domains - the bidisc, the symmetrized bidisc and the annulus - which are affiliated to given kernels.