D. Voiculescu introduced two main notions of free entropy for a given tuple of self-adjoint operators $X$: the microstates free entropy $\chi(X)$ and the non-microstates free entropy $\chi^*(X)$. In a joint work with David Jekel, we give an elementary proof of the inequality $\chi(X) \leq \chi^*(X)$ (originally proved by Biane, Capitaine, and Guionnet). Furthermore, we extend the inequality to conditional free entropy, and limits along arbitrary ultrafilters for the microstate spaces. The proof illuminates relationships between the free entropy of a tuple $X$ and the classical entropy of matrix approximations to $X$.

Bio: Jenny Pi is a current graduate student at UC Irvine, under supervision of Isaac Goldbring and Michael Cranston. Her research in free probability is related to establishing connections between differing notions of free entropy. She is interested in using free probability to study the internal structure of von Neumann algebras and to make connections with quantum groups.