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Explore a 51-minute mathematical lecture that surveys results related to q-deformations of Fock spaces and q-convolutions for probability measures on the real line R. Delve into the combinatorics of measure moments and various types of q-cumulants, with particular focus on q-convolutions connected to classical continuous and discrete q-Hermite polynomials. Learn about classical convolutions (q = 1), free and Boolean relations (q = 0), and examine new q-analogue classical convolutions developed by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. Discover the construction of Brownian motion related to q-Discrete Hermite polynomials of type I, while exploring questions about the positivity of q-convolutions. Master key concepts including orthogonal polynomials, measures convolution, Khintchine inequality, and q-Gaussian operators in this advanced mathematics presentation delivered by Marek Bożejko from Wroclaw University at the Institut des Hautes Etudes Scientifiques.
