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Independence and Convolutions in Noncommutative Probability

Noncommutative probability, also called quantum probability or algebraic probability theory, is an extension of classical probability theory where the algebra of random variables is replaced by a possibly noncommutative algebra. A... ver más

31/03/2015

UFC

194K€

Presupuesto del proyecto: 194K€

Líder del proyecto

UNIVERSITE DE FRANCHECOMTE No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Fecha límite participación Sin fecha límite de participación.

Financiación concedida El organismo FP7 notifico la concesión del proyecto el día 2015-03-31 No tenemos la información de la convocatoria

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Líder del proyecto

UNIVERSITE DE FRANCHECOMTE No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 194K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto Noncommutative probability, also called quantum probability or algebraic probability theory, is an extension of classical probability theory where the algebra of random variables is replaced by a possibly noncommutative algebra. A surprising feature of noncommutative probability is the existence of many very different notions of independence. The most prominent among them is freeness or free probability, which was introduced by Voiculescu to study questions in operator algebra theory. In the last twenty-five years, free probability has turned into a very active and very competitive research area, in which analogues for many important probabilistic notions like limit theorems, infinite divisibility, and L\'evy processes have been discovered. It also turned out to be closely related to random matrix theory, which has important applications in quantum physics and telecommunication. The current project proposes to study the mathematical theory of independence in noncommutative probability, and the associated convolution products. We will concentrate on the following topics: (1) Applications of monotone independence to free probability. Some applications have been found already, but recent work indicates that much more is possible. (2) Analysis of infinitely divisible distributions in classical and free probability. 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 free independence. This concept introduces very new ideas, whose better understanding will certainly lead to new interesting results. The methods we will use in this project come not only from noncommutative probability, but also from functional analysis, complex analysis, combinatorics, classical probability, random matrices, and graph theory.

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