"Noncommutative function theory'' refers, broadly, to the study of certain functions of several noncommuting arguments (for example, polynomials in noncommuting variables) which may be thought of as a generalization of complex analysis to the noncommutative realm. (If one thinks of an ordinary analytic function as a generalized polynomial, then noncommutative ("nc'') functions generalize noncommutative polynomials.) The theory we will discuss originates with work of J.L. Taylor in the 1970's, aimed at understanding what sort of functional calculus might exist for systems of noncommuting operators. Since then it has been discovered and rediscovered by many authors, and expanded in many directions.

In these lectures we will begin with the definitions and basic properties of nc sets and nc functions, and give examples. For the subsequent development we will not try to touch on all the areas in which nc functions arise (such as free probability, free semialgebraic geometry, etc.), but instead will focus on those aspects of the theory that will be most recognizable to participants in other parts of this Focus Program. Thus, we will consider spaces of nc functions (analogous to spaces of holomorphic functions, such as the Hardy space in the disk) and examine familiar objects and problems (multipliers, reproducing kernels, interpolation theorems, inner-outer factorization, and so on) in this setting. We will try to emphasize the central role of the notion of a "realization'' in this theory, and indicate some applications of the nc machinery to problems in ordinary function spaces (in particular the Drury-Arveson space and spaces with complete Nevanlinna-Pick kernel).