## Developments with $b-$ and $q-$ finite free convolutions

#### Nick RyderUniversity of California, Berkeley (UC Berkeley)

When dealing with expected characteristic polynomials, the finite free convolutions (classically known as the Walsh and Grace-Szego Convolutions) play a central role. In a unpublished paper of Marcus, Speilman, Srivastava, they explore a random matrix interpretation of these polynomial convolutions and develop univariate methods for obtaining precise information on the movement of the max root under the convolution. Recently, some have explored related convolutions which we call the $q-$multiplicative and $b-$additive convolutions. In this talk we develop a general framework which transfers results from the multiplicative to the additive convolution, and use this to solve a recent conjecture of Branden, Shapiro, and Krasikov relating to root separation preservation under the $b-$additive convolution. This comes from joint work with Jonathan Leake.

Back to Workshop I: Expected Characteristic Polynomial Techniques and Applications