The field of Free Probability was first started in the 1980’s by Dan-Virgil Voiculescu. He was investigating a property called freeness in the context of operator algebras. Eventually these ideas developed in a way to study non-commutative probability spaces. This later lead to connections with random matrices. Properties and concepts in Free Probability often have direct analogues with those in classical probability. We will begin with developing the notation of free independence which corresponds to independence in classical probability theory.

In the 1990s, Terao and Yuzvinsky conjectured that reduced hyperplane arrangements satisfy the Logarithmic Comparison Theorem, asserting that the logarithmic de Rham complex computes the cohomology of the arrangement's complement. Essentially, this replaces the Brieskorn algebra in Brieskorn's Theorem with the logarithmic de Rham complex. We prove this conjecture by, among other things, sharply bounding the Castelnuovo--Mumford regularity of logarithmic j-forms of a central, essential, reduced arrangement. Time permitting we will discuss how to extend this untwisted Logarithmic Comparison Theorem to a twisted version. Here the twisted logarithmic de Rham complex computes the cohomology of the arrangement's complement with coefficients the rank one local system corresponding to the twist. Unlike the twisted Orlik--Solomon algebra, which can only computes a subset of the rank one local systems on the complement, this generalization computes all such rank one local systems.

In this joint work with Joaquin Rodrigues Jacinto, we develop the theory of locally analytic representations from the perspective of Condensed mathematics of Clausen-Scholze. Taking as inspiration foundational works on the subject from Lazard/Schneider - Tetelbaum/Emerton, etc., we reprove and generalize some cohomological comparisons between continuous, locally analytic and Lie algebra cohomology.