Approximation, interpolation, and lifting on the unit ball

We solve the Nevanlinna-Pick interpolation problem on the open unit ball of $\mathbb{C}^n$. Our solutions signify the role of inner functions on the unit ball, objects whose existence was once in doubt, and to which Aleksandrov, Rudin, Sibony, and others made fundamental contributions. The results also reveal the importance of extremal functions, which emerge as natural analogues of finite Blaschke products in the unit ball. This viewpoint is illustrated by a Carath\'{e}odory approximation theorem and a unit ball analogue of Pick's theorem: every solvable interpolation problem admits an extremal function solution. We also solve the commutant lifting problem, where both inner functions and extremal functions play a fundamental role. These results resolve several well-known problems on the unit ball.