Professor Rovnyak will pursue developments consequent upon the proof of the Bieberbach conjecture in complex analysis. The latter has led to a study of contractive substitution trans- formations in certain Krein spaces that include and generalize the Dirichlet space. Properties of these transformations will be explored both from the point of view of invariant subspace theory and the theory of linear operators on spaces with indefinite metric. A long-range goal is a coefficient body problem that seeks to characterize initial segments of power series which represent normalized Riemann mapping functions that map the disk into itself. Related abstract extension problems and commutant lifting theorems for contraction operators on Krein spaces will also be studied. The research envisioned here is about analytic functions, a central preoccupation of mathematics for well over a century. They can be defined variously as solutions of a certain simple system of partial differential equations, as maps which take planar regions to other planar regions in a way that preserves angles except at isolated singularities, or as limits of polynomials in a suitably precise sense. Corresponding to the diverse ways of describing them, analytic functions can be studied in a variety of ways. The approach favored in this project is operator-theoretic, building up Hilbert spaces of functions by taking polynomials and their limits in a particular fashion, then studying operators on the spaces that arise naturally from the functions, e.g. by multiplication or by composition. Conversely, these function-theoretic operators often turn out to be quite general in operator theory, so by studying the functions one can learn a great deal about operators in the abstract.