Eli Shamovich, Pure Mathematics, University of Waterloo
"Polynomials and rational functions"
In this talk, we will apply the theory of noncommutative function to two classes of examples, namely polynomials and rational functions. Polynomials for us are elements of the free algebra, but what are rational functions? We will show that an nc function given on each level by matrices of polynomials in the coordinates of uniformly bounded degree is, in fact, an element of the free algebra. We will define what are rational nc functions using ideas of P. M. Cohn and Amitsur and discuss their realization theory. This will allow us to discuss various results in free probability, algebra and even the theory of computations. In particular, we will discuss the result of Helton, Mai, and Speicher that allows one to evaluate rational functions on stably finite algebras and its application to random matrices.