Components of the Research supported by the ERC Advanced Grant NCDFP
The first project deals with quantum symmetries of non-commutative distributions. We try to classify non-commutative symmetries and describe the effect of invariance under such quantum symmetries for non-commutative distributions. This is based on the theory of easy quantum groups.
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.
- Banica, Curran, Speicher: Stochastic aspects of easy quantum groups, Probab. Theory Related Fields 149, 435-462, 2011.
- Banica, Curran, Speicher: De Finetti theorems for easy quantum groups, Ann. Probab. 40 (1), 401-435, 2012.
- Banica, Speicher: Liberation of orthogonal Lie groups, Adv. Math. 222 (4), 1461-1501, 2009.
- Köstler, Speicher: A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation, Commun. Math. Phys. 291 (2), 473-490, 2009.
- Raum, Weber: The full classification of orthogonal easy quantum groups, arXiv:1312.3857, 2013.
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.
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.
- Biane, Speicher: Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Prob. Theory Related Fields 112, 373-410, 1998.
- Biane, Speicher: Free diffusions, free entropy and free Fisher information, Annales de l'IHP 37, 581-606, 2001.
- Guionnet, Shlyakhenko: Free diffusions and matrix models with strictly convex interaction, Geom. Funct. Anal. 18, 1875.1916, 2009.
- Kemp, Nourdin, Peccati, Speicher: Wigner chaos and the fourth moment, Ann. Prob. 40, 1577-1635, 2012.
Analytic Aspects of Operator-Valued Free Probability
In the third project, we will study the analytic theory of operator valued free convolutions. One specific goal in this context is to find and implement algorithms for calculating non-commutative distributions and asymptotic eigenvalue distributions for general random matrix problems.
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.
- Belinschi, Bercovici: A new approach to subordination results in free probability, Journal d'Analyse Mathematique 101, 357-365, 2007.
- Belinschi, Mai, Speicher: Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem, arXiv:1303.3196, 2013.
- Belinschi, Speicher, Treilhard, Vargas: Operator-valued free multiplicative convolution: analytic subordination theory and applications to random matrix theory, arXiv:1209.3508, 2012.
- Speicher: Combinatorial theory of the free product with amalgamation and operator-valued free probability, Memoirs of the AMS 132, x+88, 1998.
- Voiculescu: Aspects of free analysis, Jpn. J. Math. 3, 163-183, 2008.
Department of Mathematics
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