An introduction to free analysis

The study of multivariate analytic functions in matrix or operator variables goes back to the seminal work of Taylor in the 1970s. Revived by Voiculescu in the 00s, free analysis or noncommutative function theory is now quickly developing, and its advances are fueled by questions in free probability, control theory, operator theory, matrix convexity and free real algebraic geometry.

With no prerequisite knowledge needed, this talk overviews the very basics of free analysis: noncommutative functions, topologies on noncommutative space, noncommutative differentials, Taylor–Taylor series, and finally the fundamental theorem connecting all of them — a noncommutative function is locally bounded if and only if it is analytic.