In the 1980s, L. Brown introduced an analogue of eigenvalue distribution in the framework of operator algebras, now called Brown measure. I will discuss main ingredients in some recent works on the Brown measure of a sum of two free random variables, one of which is circular, elliptic, or R-diagonal. It is shown that subordination functions that appear in the study of free additive convolution can detect information about the Brown measure. In many cases, this leads to an explicit calculation of Brown measures. These Brown measures are related to the limiting eigenvalue distributions of the full rank perturbations of i.i.d., Wigner, elliptic, and single ring random matrix models. Some ideas were used implicitly in earlier works of Dykema, Haagerup, Schultz and their coauthors, as well as some physics papers. Later, Belinschi-Sniady-Speicher made the method of Hermitian reduction and subordination be transparent in the operator-valued framework. I will explain how this method works for our models. The talk is based on my recent works arXiv:2108.09844, arXiv:2209.11823 (joint with Serban Belinschi and Zhi Yin), and arXiv:2209.12379 (joint with Hari Bercovici).

Bio: Ping Zhong has been working at the University of Wyoming since 2018. He earned his doctorate from Indiana University Bloomington in 2014, under the supervision of Hari Bercovici. He is interested in free probability and its interactions with operator algebras and random matrix theory. His recent research focuses on analytic theory of free probability, Brown measure, non-Hermitian random matrices, and their applications.