Abstract:

Roughly (and incompletely) speaking, complex analysis is based on the theory of polynomials. In a very analogous fashion, free analysis is based on the functions one gets by evaluating elements of the free algebra in some number of variables on tuples of not-necessarily commuting matrices.  This analogy lies at the heart of a rapidly growing subject which makes contacts with numerous areas of mathematics, including (but not limited to) operator algebra, complex and real algebraic geometry, and optimization theory. The talks in the OT seminar that I will give describe some of the work I have done in the subject, work which leads to the identification of the "natural habitat" of free functions.