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Independence and Convolutions in Noncommutative Probability (ICNCP)
Start date: Apr 1, 2013, End date: Mar 31, 2015 PROJECT  FINISHED 

"Noncommutative probability, also called quantum probability or algebraicprobability theory, is an extension of classical probability theory where thealgebra of random variables is replaced by a possibly noncommutativealgebra. A surprising feature of noncommutative probability is the existenceof many very different notions of independence. The most prominent among themis freeness or free probability, which was introduced by Voiculescu to studyquestions in operator algebra theory. In the last twenty-five years, freeprobability has turned into a very active and very competitive research area,in which analogues for many important probabilistic notions like limittheorems, infinite divisibility, and L\'evy processes have been discovered. Italso turned out to be closely related to random matrix theory, which hasimportant applications in quantum physics and telecommunication.The current project proposes to study the mathematical theory of independencein noncommutative probability, and the associated convolution products. Wewill concentrate on the following topics:(1) Applications of monotone independence to free probability. Someapplications have been found already, but recent work indicates that much moreis possible.(2) Analysis of infinitely divisible distributions in classical and freeprobability. Common complex analysis methods will be used for both classes,and we expect more insight into their mutual relations.(3) Application and development of Lenczewski's matricial freeindependence. This concept introduces very new ideas, whose betterunderstanding will certainly lead to new interesting results.The methods we will use in this project come not only from noncommutativeprobability, but also from functional analysis, complex analysis, combinatorics, classical probability, random matrices, and graph theory."

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