Non-commutative Probability Theory for Topological Data Analysis

Recent developments have found unexpected connections between non-commutative probability theory and algebraic topology. In particular, Boolean cumulants functionals seem to be important for describing morphisms of homotopy operadic algebras. We provide new elementary examples which clearly resemble a connection between algebraic topology and non-commutative probability, based on spectral graph theory. These observations are important for bringing new ideas from non-commutative probability into TDA and stochastic topology, and in the opposite direction.