Description of the ERC Advanced Grant NCDFP
Type of grant: |
ERC Advanced Grant |
Title: |
Non-Commutative Distributions in Free Probability |
Funded by: |
European Research Council |
Principal Investigator: |
Roland Speicher |
Duration: |
1 February 2014 - 31 January 2019 |
Aim: |
The aim of the grant is to study new directions in free probability theory with
high potential to lead to breakthroughs in our understanding of random matrix models
and operator algebras. We will drive forward the study of free analysis which is intended to provide a whole new mathematical theory for variables with the highest degree of non-commutativity and which lies at the crossroad of many exciting mathematical subjects. More specifically, the objective of the research founded by this grant is to extend our armory for dealing with non-commutative distributions and to attack some of the fundamental problems which are related to such distributions, like: the existence and properties of the limit of multi-matrix models; the isomorphism problem for free group factors, and more generally, properties of free entropy and free entropy dimension as invariants for von Neumann algebras. |
Components of the research supported by the ERC Advanced Grant NCDFP
Details: (hide the details) Looking on quantum groups and their actions has been a common theme in many investigations in mathematics in the last decades. However, in most cases one considered quantum groups which are deformations of classical groups. On contrast, the class of the so called easy quantum groups are not given by deformations, but by strengthenings of classical groups - thus they correspond to stronger kinds of quantum symmetries than their classical counter parts. More specifically, they are determined - via their intertwiner spaces - by the combinatorics of partitions. Apart from the, by now quite well-understood, quantum versions of permutation or orthogonal groups (by Wang) there is a whole new universe of quantum versions of the classical hyperoctahedral group (the latter describes the symmetries of a hypercube). These should correspond to different quantum symmetries for non-commutative distributions, see for instance the non-commutative de Finetti theorems. The exploration of those quantum symmetries promises a much deeper understanding of the non-commutative world and should also result in new types of non-commutative distributions. Literature:
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Free Malliavin Calculus: |
In the second project we will develop the theory of free Malliavin calculus. This will then be used to investigate regularity properties of non-commutative distributions. |
Details: (hide the details) One way to deal with non-commuting variables x1 , ... , xn is to try to understand suffciently many functions f(x1 , ... , xn) of them. We are faced with the question: What can we say about the distribution of the variable f(x1 , ... , xn), given the non-commutative distribution of x1 , ... , xn? Even if we take very elementary non-commutative distributions for x1 , ... , xn and polynomials as functions f, this is a hard question and not much is known. In the classical case one way to attack such questions is the use of classical Malliavin calculus. This is a very powerful tool for deriving regularity properties of solutions of stochastic differential equations. We plan to develop a non-commutative counterpart of this theory, allowing to deal with the above mentioned kind of regularity questions for non-commutative distributions. A basis of a free Malliavin calculus has been established by Philippe Biane and Roland Speicher, but it needs to be extended to deal with more sophisticated analytic questions. Such regularity questions are in particular of importance for addressing questions about free entropy and free entropy dimension, which are intimately connected with some of the most challenging and interesting problems in free probability, operator algebras, and random matrices. Literature:
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Details: (hide the details) Operator-valued free probability theory is a generalization of the usual (scalar-valued) free probability theory. Because of this generality it has a much wider range of applicability. For more advanced questions however we need a better understanding of the analytic properties of this theory, in particular of their computational aspects. It has become increasingly clear in recent years that the subordination formulation of free convolution is the most promising route to success and an extension of this to the operator-valued situation is needed. First results in this direction were recently achieved in work of Belinschi, Speicher, Treilhard, Vargas and Mai. The consequences of these ideas have to be investigated, both on a qualitative and a quantitative level. Furthermore, these results need to be extended to non-selfadjoint situations. Literature:
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Updated: 13 September 2013 Moritz Weber | Impressum |