Centre for Mathematical Physics and Stochastics

Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

Sandbjerg Estate, Denmark

Random matrices have been of great importance in statistics since the work
of J. Wishart (1928) and P.L. Hsu (1939).

In the 1950's E.P. Wigner used random matrices in an attempt to model the energy levels at nuclear reactions in quantum physics. He discovered, in particular, that the spectral distribution of a selfadjoint (Gaussian) n-by-n random matrix is, asymptotically (as n tends to infinity), the semi-circle distribution. Since then, random matrices have played a significant role in various fields of mathematics and physics.

In the 1980's, D.V. Voiculescu introduced the notion of free independence, and he proved that independent selfadjoint (Gaussian) n-by-n random matrices are asymptotically (as n tends to infinity) freely independent. This discovery has lead to a number of important applications of random matrices in operator algebra theory.

Another breakthrough in the theory of random matrices came in the 1990's, when C. Tracy and H. Widom found, for any fixed number k, the asymptotic (as n tends to infinity) joint distribution of the k largest eigenvalues of a selfadjoint Gaussian n-by-n random matrix (in the real, complex and quaternionic cases). These results have had great impact on the theory of longest increasing subsequences for random permutations.

Random matrices are also of great importance in analytical number theory,
in particular for problems concerning the distribution of zeroes for the
Riemann zeta-function and the Dirichlet L-series. This fact is
well-documented in the monograph *Random matrices, Frobenius eigenvalues,
and monodromy* by P. Sarnak and N.M. Katz.

Last, but not least, random matrices play a crucial role in many applied areas as well. We mention, in particular, areas like signal processing and neural networks which have been studied extensively by U. Grenander, J.W. Silverstein and others by virtue of random matrices.

The aim of the workshop is to bring together scientists working with the various aspects of random matrix theory, with the purpose of presenting the state of the art in each of these fields and illuminating the interrelations.

The workshop will be held at Sandbjerg Manor, a conference centre owned by the University of Aarhus, situated in the southern part of Jutland, Denmark.

Uffe Haagerup, University of Southern Denmark Odense

Steen Thorbjørnsen, University of Southern Denmark Odense

**Monday, January 13**

09:00-10:00: Dan Voiculescu: Free probability topics

10:00-10:25: Coffee/tea

10:25-11:10: Dimitri Shlyaktenko: On Voiculescu's non-microstates free entropy

11:15-12:00: Alexandru Nica: Annular non-crossing permutations, and
second-order asymptotics for random matrices

12:15-13.15: Lunch

15:00-15:30: Coffee/tea

15:30-16:15: Fumio Hiai: Free pressure and random matrix perturbations

16:20-17:05: Denes Petz: TBA

17:15-18:00: Marek Bozejko: Generalized Brownian motions associated with
Thoma characters of infinite permutation groups

18:00: Dinner

20:30: Get-together

**Tuesday, January 14**

09:00-10:00: Hari Bercovici: Eigenvalue inequalities in a von Neumann algebra

10:00-10:25: Coffee/tea

10:25-11:10: Jack Silverstein: CLT for linear spectral statistics of large
dimensional sample covariance matrices

11:15-12:00: Zhidong Bai: On the convergence of the spectral empirical
process of Wigner matrices

12:15-13.15: Lunch

15:00-15:30: Coffee/tea

15:30-16:15: Gilles Pisier: Applications of matrix models to the study of
operator spaces

16:20-17:05: Gérard Letac: The explicit computation of the moments of the Wishart distribution

17:15-18:00: Mireille Capitaine and Muriel Cassalis: Asymptotic freeness by
generalized moments. Application to Wishart and Beta matrices

18:00: Dinner

**Wednesday, January 15**

09:00-10:00: Alexander Gamburd: Expander graphs, random matrices, and quantum chaos

10:00-10:30: Coffee/tea

10:30-11:15: Nina Snaith: Random matrix theory and the Riemann zeta function

11:20-11:50: Francesco Mezzadri: Random matrix theory and the zeros of zeta(s)

12:15-13.15: Lunch

13:30-17.30: Excursion

18:00: Dinner

**Thursday, January 16**

09:00-10:00: Alexander Soshnikov: On Janossy densities in determinantal and pfaffian ensembles

10:00-10:25: Coffee/tea

10:25-11:10: Boris Khoruzhenko: Random circulant Jacobi matrices

11:15-12:00: Romuald Janik: Explicit factorization of Seiberg-Witten curves with matter from random matrix models

12:15-13.15: Lunch

15:00-15:30: Coffee/tea

15:30-16:15: Jan Ambjørn: A matrix model description of 3d quantum gravity.

16:20-16:50: Franz Lehner: Noncommutative cummulants and characterizations
of generalized Gaussian random variables

17:10-17:40: Arno Kuijlaars: Density of states in a 2-matrix model

17:45-18:15: Madalin Guta: Sieve maximum likelihood estimation of the density
matrix through quantum demography

19:00: Conference dinner

**Friday, January 17**

09:00-10:00: Anatoly Vershik: Fock factorizations and the processes with independent values

10:00-10:25: Coffee

10:25-11:10: Stanislaw Szarek: TBA

11:15-12:00: Benoit Collins: Itzykson-Zuber type integrals and free probability

12:15-13.15: Lunch

13.15-14.00: Piotr Sniady: Non-hermitian operators, random matrices and Brown spectral measure

14.05-14.50: Ken Dykema: Invariant subspaces for operators modeled by
certain upper triangular random matrices

No financial support is available. Participants are expected to have their expenses covered by their home institutions.

The registration fee is DKK 500, payable upon arrival at Sandbjerg in cash in
Danish currency.

Sandbjerg is located in South Jutland, close to the German border.

Address: Sandbjerg Manor, Sandbjergvej 102,
DK-6400 Sønderborg

**By airplane:** *Flights from Copenhagen Airport to
Sønderborg*: 8:00-8:40, 11:00-11.40, 14:55-15:35 (Monday-Friday); 17:15-17:55 (Monday-Friday and Sunday); 20:15-20:55 (Wednesday-Sunday); 18:15-18:55 (Monday-Friday and Sunday).

*Flights from Sønderborg to Copenhagen Airport*: 7:00-7:40
(Monday-Friday); 7:45-8:25 (Saturday); 9:05-9:45 (Monday-Friday);
12:30-13:10 (Monday-Friday); 15:55-16:35 (Monday-Friday and Sunday);
18:15-18:55 (Monday-Friday and Sunday).

Distance between the airport and Sandbjerg Estate approx.12 km.

**By train:** *Direct train from the airport and from the
central railway
station of Copenhagen to Sønderborg*: Every second hour from 5:30
in the morning to 21:30 in the evening. The travel time is approx. 3 hours and
45 minutes.

*Direct train from Sønderborg to the central railway station of
Copenhagen*: Every second hour from 6:00 in the morning to 20:00 in the
evening. The travel time is approx. 3 hours and 45 minutes.

Distance between the railway station and Sandbjerg Estate approx. 7 km.

**By car:**

From the E45 southbound:

Take exit 71 (Kruså) towards Sønderborg (road no. 41).
In Vester Sottrup, 7.6 km before Sønderborg, a sign
on the left reads: Sandbjerg 3 km.

From the E45 northbound:

Take exit 75 (Kruså) towards Sønderborg (road no. 8).
Driva via Kruså, Rinkenæs and Egernsund to Broager.
From here, turn north towards Nybøl and Vester Sottrup where
there is a sign which reads: Sandbjerg 3 km.

See the information map.

`maphysto@maphysto.dk`

) or to the organizers Ole E. Barndorff-Nielsen (`oebn@imf.au.dk`

), Uffe Haagerup (`haagerup@imada.sdu.dk`

), and Steen Thorbjørnsen (`steenth@imada.sdu.dk`

).

(This announcement is also available in [ postscript-format | pdf-format ])

This document, *http://www.maphysto.dk/oldpages/events/RandMat2003/index.html*,
was last modified
January 19, 2004.
Questions or comments to the contents of this document should
be directed to `maphysto@maphysto.dk`.