Ph.D. (University of Heidelberg)

Habilitation in Mathematics (University of Heidelberg)

I graduated in physics with a work on chaotic systems. In particular, I made experimental and numerical studies on the chaotic behaviour of heart cells. Although my interest has in the meantime shifted to more mathematical subjects, I always dream of finding some applications of mathematical theories to physical or biophysical problems.

Starting with my PhD I have been involved in mathematical investigations in the field of non-commutative probability theory, a hybrid of functional analysis and probability theory. In particular, I am fascinated by one of the jewels of that theory, namely by the so-called free probability theory. This theory was introduced and developed by Dan Voiculescu as a tool for studying the structure of a special class of operator algebras (namely von Neumann algebras coming from free products of groups). And indeed this Theory has given a lot of new results for operator algebras (especially concerning the von Neumann algebras of free groups). This progress, however, has only been possible because surprising relations between free probability theory and quite different fields of mathematics and physics have emerged.

My own work concentrates on these relations of free probability theory with other fields. In particular, I am studying the combinatorial and probabilistic aspects of free probability and also its applications to problems in quantum statistical physics.

For example, I developed a free stochastic analysis for the free Brownian motion and investigated its relation with random matrices, I showed that the combinatorial side of free probability is governed by the lattice of non-crossing partitions, I derived a master-equation for the description of memory effects in open systems and I considered deformations of the canonical commutation and anti-commutation relations.

Combinatorics of free cumulants (with B. Krawczyk), J. Comb. Th., Ser. A 90 (2000), 267-292.

Free diffusions, free entropy and free Fisher information (with P. Biane), Preprint, 2000.

Free probability theory and free diffusion, Preprint 2000 (Proceedings of joint German-Japanese workshop in Kyoto).

Physical applications of freeness (with P. Neu), Proceedings of 'XIIth International Congress of Mathematical Physics' (ICMP '97, Brisbane), International Press, 1999, 261-266.

Some minimization problems for the free analogue of the Fisher information (with A. Nica and D. Shlyakhtenko), Advances in Mathematics 141 (1999) 282-321.

Commutators of free variables (with A. Nica), Duke Math. J. 92 (1998) 553-592.

Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Memoir of the AMS 627 (1998).

Stochastic calculus with respect to free Brownian motion and analysis on Wigner space (with P. Biane), Prob. Th. Rel. Fields 112 (1998) 373-410.

Free calculus, Preprint 1998 (summer school on 'Quantum Probability', Grenoble, 1998).

Free probability theory and non-crossing partitions, Seminaire Lotharingien de Combinatoire B39c (1997).

q-Gaussian processes: non-commutative and classical aspects (with M. Bozejko and B. Kummerer), Commun. Math. Phys. 185 (1997) 129-154.

Rigorous mean-field theory for Coherent Potential Approximation: Anderson model with free random variables (with P. Neu), J. Stat. Phys. 80 (1995) 1279-1308.

On the multiplication of free n-tuples of non-commutative random variables (with A. Nica), Amer. J. Math. 118 (1995) 799-837.

Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann. 298 (1994) 611-628.

A self-consistent master equation and a new kind of cumulants (with P. Neu), Z. Phys. B 92 (1993) 399-407

An example of a generalized Brownian motion (with M. Bozejko), Commun. Math. Phys. 137 (1991) 519-531.

A new example of 'independence' and 'white noise'. Prob. Th. Rel. Fields 84 (1990) 141-159.

Regular and chaotic behaviour of cardiac cells stimulated at frequencies between 2 and 20 Hz (with J. Hescheler), Eur. Biophys. J. 17 (1989) 273-280.

The free probability group at Queens consists of myself, Jamie Mingo (operator algebras), and Claus Koestler (non-commutative probability theory), a post-doctoral fellow. We also have close ties with Alexandru Nica of the University of Waterloo. As a new member of the Department of Mathematics and Statistics at Queens I am looking forward to working with graduate students who share our enthusiasm for free probability theory.