and Integrable Systems

on the campus of the Université de Montréal.

Jacques Hurtubise

Participants

This program tried to emphasize the remarkable connections between two domains that a priori seem unrelated: random matrices (together with associated random processes) and integrable systems. The relations between random matrix models and the theory of classical integrable systems have long been studied. These appear mainly in the deformation theory, when parameters characterizing the measures or the domain of localization of the eigenvalues are varied. The resulting differential equations determining the partition function and correlation functions are, remarkably, of the same type as certain equations appearing in the theory of integrable systems. They may be analyzed effectively through methods based upon the Riemann-Hilbert problem of analytic function theory and by related approaches to the study of nonlinear asymptotics in the large N limit. Associated to studies of matrix models are certain stochastic processes, the "Dyson processes", and their continuum limits, which are related to the spectra of random matrix ensembles, and may also be studied by related methods.

Besides the well-known physical applications of random matrix theory, such as the Wigner-Dyson statistical approach to the distribution of high lying resonances of large nuclei, and the more recent applications to string theory and two dimensional quantum gravity, there exist further new applications under current study, such as the computation of correlation functions in supersymmetric Yang-Mills theory, and the regularization of the Laplacian growth problem of two dimensional fluid dynamics. Correlation functions between eigenvalues of random matrices also have close similarities to those in integrable quantum spin systems and many body models. There are further remarkable connections to a variety of probabilistic problems such as random words, tilings and partitions, as well as to the statistical distribution of zeros of L functions.

The program provided an opportunity for productive interactions, bringing together top experts and younger researchers beginning work in this area. The schedule consisted of two parts. There were eight extended lecture series on related topics, each of one week's duration, having a survey and pedagogical character, aimed primarily at younger researchers entering the field. The afternoon sessions were of "workshop" character, with one hour talks presented on current work in the field.

- Spectral theory of random matrices
- Determinantal ensembles
- Integral operators in random matrix theory
- Dyson processes. Airy, Bessel, sine and Laguerre processes
- Matrix Riemann-Hilbert methods, applications to large N asymptotics
- Differential equations for gap distributions and transition probabilities
- Relations to integrable systems and isomonodromic deformations
- Growth processes; applications to fluid dynamics and crystal growth
- Applications to random tilings, random words, random partitions
- Applications to L-functions
- Applications to multivariate statistics

## Lecture series speakers | ||

Mark Adler | Brandeis Univ. | |

Pavel Bleher |
Indiana Univ.-Purdue Univ. at Indianapolis (I.U.P.U.I.) | |

Bertrand Eynard |
C.E.A,. Saclay, S.Ph.T. | |

Alexander Its | Indiana
Univ.-Purdue Univ. at Indianapolis (I.U.P.U.I.) | |

Ken McLaughlin | University of Arizona | |

Craig Tracy |
U.C. Davis | |

Pierre van Moerbeke |
Univ. Catholique de Louvain / Brandeis Univ. | |

Harold Widom |
U.C. Santa Cruz | |

## Workshop speakers |
||

Marco Bertola |
Concordia Univ., C.R.M. | |

Brian Conrey |
American Institute of Mathematics | |

Percy Deift |
Courant Inst., N.Y.U. | |

Philippe di Francesco |
C.E.A., Saclay, S.Ph.T. | |

Sam Howison |
Math. Institute, OCIAM, Oxford | |

Vladimir Kazakov |
E.N.S., Paris | |

Dmitri Korotkin |
Concordia Univ., C.R.M. | |

Arno Kuijlaars |
Katholieke Universiteit Leuven | |

Andrei Okounkov |
Princeton Univ. | |

Alexander Orlov |
Oceanology Inst., Moscow | |

Alexander Soshnikov |
U.C. Davis | |

Nina Snaith |
University of Bristol | |

Anton Zabrodin |
I.T.E.P., Moscow | |

Ofer Zeitouni |
Univ. Minn., Technion | |

Paul Zinn-Justin |
Orsay, Univ. Paris-sud | |

Jean-Bernard Zuber |
C.E.A., Saclay, S.Ph.T. |

A refereed Special Issue of Journal of Physics A: Mathematical and General was published in July, 2006 based on the topics covered in the workshop part program