**Foundations and Applications of Random Matrix Theory in Mathematics and Physics**

Organized by Alexei Borodin,Peter Forrester, Yan Fyodorov, Alice Guionnet, Jon Keating, Mario Kieburg, and Jacobus Verbaarschot

*August 24 – December 18, 2015*

Weekly Talks are held in SCGP Room 313 at 11:00 am.

Random Matrix theory has been applied to many areas in pure and applied mathematics and in physics, ranging from correlations among the zeros of the Riemann function and the distribution of the longest increasing subsequences of permutations to the spacing distribution of nuclear levels and correlations of the eigenvalues of the Dirac operator in Quantum Chromo Dynamics. In this program we will discuss recent developments of random matrix theories beyond the ten fold classification in terms of large symmetric spaces. In particular we will focus on the following four areas: 1) Chiral random matrix theories with applications to gauge theories and the Riemann function. Nonperturbative properties of strongly interacting quantum held theories can be understood by means of random matrix theories with the same chiral symmetry breaking pattern. Recent work indicates that the same is true if the symmetries are softly broken by discretization effects or a chemical potential. Applications to the zeros of the Riemann function, Dirac spectra in lower dimensions and relations with topological insulators will be discussed as well. 2) The relation between integrability and random matrix theory. Invariant random matrix theories can be solved because of underlying integrable structures such as for example the Toda lattice equation. More complicated ensembles still can be solved suggesting that an underlying integrable structure exists and we hope to explore such relations in this program. 3) Universal properties on non-invariant random matrix ensembles. Examples are Wigner matrices, ensembles of sparse matrices and band matrices. In particular, band matrices describe the transition between Poisson and Wigner-Dyson eigenvalue statistics and have been used to study scaling properties of localization. Based on recent work showing that these properties are universal we hope to make further progress in this area. 4) Random Matrix Theory and dynamics. This goes back as far as Dyson Brownian motion model for the eigenvalues of random matrices, but recently this topic received a great deal of attention in the context of non-equilibrium dynamics and the Kardar-Parisi-Zhang equation. This is also the topic of the workshop Random matrices, random growth processes and statistical physics from September 6-11 which is part of this program. Related issues such as the Langevin evolution of random matrix spectra and relations between the stochastic Loewner equation and random matrix theory will be discussed as well. A second workshop related to physics applications of random matrix theory may be organized later in the program.

Program application is now closed.

Date | Speaker | Title |

8/28 | Peter Forrester | Raney distribution and random matrices |

9/14 | Shinsuke Nishigaki, Shimane University, Japan | Tracy-Widom distribution and spontaneous SUSY breaking in a matrix model of 2D IIA superstrings |

9/16 | Arno Kuijlaars, Universiteit van Leuven, Belgium | Products of random matrices |

9/18 | Pierre van Moebeke University of Louvain, Belgium | The Tacnode GUE-Minor Process |

9/30 | Mariya Shcherbina, Institute for Low Temperature Physics, National Ac. Sci. of Ukraine |
Central limit theorem for linear eigenvalue statistics of random matrices with independent or weakly dependent entries. |

10/1 | Hans Weidenmueller, Max Planck Institut for Kernphysik, Heidelberg | Neutron Resonance Widths and the Porter-Thomas Distribution |

10/5 | Tilo Wettig, University of Regensburg | On Zirnbauer’s approach to induced QCD |

10/7 | Mark Adler, Brandeis University | Double Aztec Diamonds and Lozenge Tilings with Irregular Boundaries |

10/9 | Gernot Akemann, Bielefeld University | Recent progress in products of random matrices |

10/12 | Francesco Mezzadri (Bristol University) | Global Fluctuations of Linear Statistics of beta Ensembles |

10/14 | Nina Snaith (Bristol University) | Combining random matrix theory and number theory |

10/16 | Tatyana Shcherbina (St. Petersburg State University) | Random band matrices: delocalization and universality |

10/23 | Thomas Seligman, UNAM Mexico and CIC Cuernavaca | Random matrix ensembles of density matrices from first principles and from random matrix dynamics. [1] |

10/28 | Manan Vyas | Random Matrix Theory for Quantum Many-body systems |

10/30 | Fabio Franchini, ICTP Trieste | Ergodicity breaking in invariant matrix models |

11/9 | Mario Kieburg, University of Duisburg-Essen | Products of Random Matrices and Quantum Information Theory |

11/11 | Pjotr Warchot, Jagiellonian University, Cracow | Correlators of left and right eigenvectors of non-Hermitian random matrices. |

11/13 | James Osborn, Argonne National Laboratory | Lattice Dirac Fermions and Chiral Effective Theories |

11/18 | Miguel Tierz, University of Madrid | Matrix models in Chern-Simons-matter theory |

11/20 | Nivedita Deo, University of Delhi | Counting RNA Folds From Random Matrix Models And Networks |