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Explore a 38-minute lecture from the Free Entropy Theory and Random Matrices Workshop at UCLA's Institute for Pure & Applied Mathematics, where Jennifer Pi from UC Irvine presents groundbreaking research on relating free entropic quantities. Delve into the comparison between Voiculescu's two notions of free entropy for self-adjoint operator tuples - the microstates free entropy χ(X) and non-microstates free entropy χ∗(X). Learn about the elementary proof developed by Pi and David Jekel demonstrating the inequality χ(X)≤χ∗(X), originally established by Biane, Capitaine, and Guionnet. Discover how this proof extends to conditional free entropy when conditioning upon any separable von Neumann subalgebra, and understand the fundamental connections between free entropy of a tuple X and the classical entropy of its matrix approximations.
