**ORGANIZERS:**

Brian Conrey, *American Institute of Mathematics*

Jon Keating, *University of Oxford*

Hugh Montgomery, *University of Michigan*

Kannan Soundararajan, *Stanford University*

**Summary:**

On April 6, 1972 a young graduate student named Hugh Montgomery and the world-renowned mathematical physicist Freeman Dyson had a conversation in the tearoom at the Institute for Advanced Study which led to a fusion of two disparate fields and an intellectual revolution that is stronger than ever today. Montgomery and Dyson discovered that the zeros of the Riemann zeta-function, important for understanding prime numbers, seem to obey the same distribution patterns as the eigenvalues of large random unitary (or hermitian) matrices, which had been extensively studied by physicists, especially to model the interactions within large atomic nuclei. This stunning connection has held up to extensive numerical and theoretical tests over the intervening years and has been extended to give random matrix models for the low lying zeros of families of L-functions and for the moments and distribution of values of the L-functions in these families and to analogous families over function fields. This conference will bring together 130 mathematical scientists to a meeting that will highlight the progress and the state of the art of results at the nexus of these two fields and will underline the questions most pertinent for further research.

**SPEAKERS:**

Louis-Pierre Arguin, City University of New York | Nina Holden, ETH Zurich |

Emma Bailey, City University of New York | Matilde Lalin, Universite de Montreal |

Sieg Baluyot, American Institute of Mathematics | Paul Nelson, Institute for Advanced Study |

Vorrapan Chandee, Kansas State University | Kyle Pratt, University of Oxford |

Alexandra Florea, University of California at Irvine | Zeev Rudnick, Tel Aviv University |

Yan Fyodorov, King’s College London | Tom Spencer, Institute for Advanced Study |

Adam Harper, University of Warwick | Matt Young, Texas A & M University |

Kannan Soundararajan, Stanford University |

Topic: Opening Remarks and History of the math talksSpeakers: Peter Sarnak, Hugh Montgomery and Jon KeatingRecording: Opening Remarks and History of the math talksAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… | Topic: The Fyodorov-Hiary-Keating ConjectureSpeaker: Louis-Pierre Arguin – City University of New YorkRecording: The Fyodorov-Hiary-Keating ConjectureAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… |

Topic: Statistical mechanics arising from random matrix theorySpeaker: Thomas Spencer – Institute for Advanced StudyRecording: Statistical mechanics arising from random matrix theoryAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… | Topic: Large deviation estimates for Selberg’s central limit theorem, applications, and numericsSpeaker: Emma Bailey – City University of New YorkRecording: Large deviation estimates for Selberg’s central limit theorem, applications, and numericsAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… |

Topic: The distribution of values of zeta and L-functionsSpeaker: Kannan Soundararajan – Stanford UniversityRecording: The distribution of values of zeta and L-functionsAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… | Topic: RMT statistics in number theory and in quantum chaosSpeaker: Zeev Rudnick – Tel Aviv UniversityRecording: RMT statistics in number theory and in quantum chaosAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… |

Topic: Large sieve inequalities for families of L-functionsSpeaker: Matt Young -Texas A&M UnviersityRecording: Large sieve inequalities for families of L-functionsAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… | Topic: Negative moments of the Riemann zeta functionSpeaker: Alexandra Florea -University of California, IrvineRecording: Negative moments of the Riemann zeta functionAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… |

Topic: Number theoretic aspects of multiplicative chaosSpeaker: Adam Harper – University of WarwickRecording: Number theoretic aspects of multiplicative chaosAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… | Topic: Half-Isolated Zeros and Zero-Density EstimatesSpeaker: Kyle Pratt -University of OxfordRecording: Half-Isolated Zeros and Zero-Density EstimatesAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… |

Topic: Gaussian multiplicative chaos: applications and recent developmentsSpeaker: Nina Holden – ETH ZurichRecording: Gaussian multiplicative chaos: applications and recent developmentsAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… | Topic: The recipe for moments of LL-functions and characteristic polynomials of random matricesSpeaker: Sieg Baluyot -American Institute of MathematicsRecording: The recipe for moments of L-functions and characteristic polynomials of random matrices Abstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… |

Topic: A few results and conjectures on some product-ratio correlation functions of characteristic polynomials of beta-Hermite ensemblesSpeaker: Yan Fyodorov – King’s College LondonRecording: A few results and conjectures on some product-ratio correlation functions of characteristic polynomials of beta-Hermite ensemblesAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… | Topic: Moments of large families of Dirichlet L-functionsSpeaker: Vorrapan Chandee – Kansas State UniversityRecording: Moments of large families of Dirichlet L-functionsAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… |

Topic: The Fyodorov-Hiary-Keating ConjectureSpeaker: Louis-Pierre Arguin – City University of New YorkRecording: The Fyodorov-Hiary-Keating ConjectureAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… | Topic: Moments and bounds for L-functions of large degreeSpeaker: Paul Nelson -Institute for Advanced StudyRecording: Moments and bounds for L-functions of large degreeAbstract URL: https://www.ias.edu/events/50-years-number-theory-and-random-matrix-the… |

**Problem Session Notes:**

**SPONSORS:****IAS, NSF, AIM, Heilbronn Institute for Mathematical Research**