Operator theory is a major discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. Subsequently, the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. Indeed, since many explicit operators arise in complex function theory, differential geometry, and approximation theory, there is a broad range of applicability. In particular, the connection between operator theory and complex function theory has long been actively pursued from various points of view. The recent brilliant innovations by deBranges that led to the solution of the Bieberbach conjecture have further developed the deep interdependence of these disciplines. Professors Rosenblum and Rovnyak form a research team that has had an excellent record of research on this interface, and their earlier work was in a distant way a precursor of deBranges' ideas. Both singly and together, their successes include seminal results in perturbation theory of unitary transformations, spectral theory of the Hilbert matrix, factorization and interpolation theory for bounded analytic functions, a generalization of the corona theorem, and extensions of the basic inequality in deBranges' work (loc cit). The current proposal seeks to extend the estimation theory of Riemann mapping functions that produced the proof of the Bieberbach conjecture. Known extensions treat powers greater than -3/2, and it is proposed to explore nonpositive integral powers. Other extensions involve interpolation theory, and have applications in operator theory. Professors Rosenblum and Rovnyk will also study interpolation theory for Newton spaces, which are finite difference analogs of Hardy spaces. Emphasis will be placed on the special functions related to Newton spaces, and to logarithmic versions of these spaces. The project also includes operator equations associated with analytic functions of bounded mean oscillation and with the corona theorem.