Random matrix approaches in number theory

Programme theme

For thirty years there have been conjectured connections, supported by ever mounting evidence, between the zeros of the Riemann zeta function and eigenvalues of random matrices. One of the most famous unsolved problems in mathematics is the Riemann hypothesis, which states that all the non-trivial zeros of the zeta function lie on a vertical line in the complex plane, called the critical line. The connection with random matrix theory is that it is believed that high up on this critical line the local correlations of the zeros of the Riemann zeta function, as well as other L-functions, are the same as those of the phases of the eigenvalues of unitary matrices of large dimension taken at random from the CUE ensemble of random matrix theory. More recently, however, it was realized that random matrix theory not only describes with high accuracy the distribution of the zeros of L-functions, but it is also extremely successful in predicting the structure of various average values of L-functions that previous number theoretic techniques had not been able to tackle.

The programme will mainly focus on how random matrix theory can further contribute to unanswered questions in number theory and on how to put the connection between random matrices and number theory on a rigorous footing. However, both random matrix theory and number theory individually play significant roles in theoretical physics and probability: random matrix statistics appear in the spectra of quantum systems whose classical limit is chaotic; the problem of quantum unique ergodicity has connections with the theory of modular surfaces and algebraic number theory; many of the main results on the statistics of ensembles of random matrices have been the work of probabilists; the Riemann zeta function even shows up in the theory Brownian motion - and this is just to name a few. These themes will also be developed through focused workshops.

The main goal of this programme is to draw on the expertise of these diverse groups to produce new ideas on how random matrix theory can tackle important problems in number theory.

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme
Random matrix theory and entanglement in quantum spin chains Authors: Francesco Mezzadri, Jon Keating Download PDF	2003	RMA
On the moments of traces of matrices of classical groups Authors: V Vasilchuk, L Pastur Download PDF	2003	RMA
The zeros of random polynomials cluster uniformly near the unit circle Authors: Ashkan Nikeghbali, Christopher Hughes Download PDF	2003	RMA
Mock-Gaussian behaviour Authors: Christopher Hughes Download PDF	2003	RMA
Symmetry classes of disordered fermions Authors: P Heinzner, A Huckleberry, Martin Zirnbauer Download PDF	2003	RMA
Explicit lower bounds on the modular degree of an elliptic curve Authors: Mark Watkins Download PDF	2003	RMA

Workshops

RMAW01

Seminars

Thursday 29th January 2004
15:00 to 16:00	Random matrices and invariant theory	Room 1
16:30 to 17:30	Small solutions to linear congruences and Hecke equidistribution	Room 1

Thursday 5th February 2004
15:00 to 16:00	Mark Watkins Pennsylvania State University Solving systems of polynomial equations via multidimensional p-adic Newton iteration	Room 1
16:30 to 17:30	Christopher Hughes Moments of the Riemann Zeta function and random matrix theory "theory"	Room 1

Thursday 19th February 2004
14:30 to 15:30	Steve Gonek University of Rochester Mean value theorems and the zeros of the zeta function	Room 1
16:00 to 17:00	David Farmer Differentiation evens out zero spacings	Room 1

Thursday 26th February 2004
16:00 to 17:00	Shin-ya Koyama Keio University Multiple zeta functions	Room 1

Thursday 4th March 2004
14:30 to 15:30	Jonathan Keating University of Bristol Negative moments	Room 1
16:00 to 17:00	Alex Gamburd Stanford University Random matrices, magic squares and the Riemann zeta function	Room 1

Thursday 11th March 2004
16:00 to 17:00	Parametric random matrix theory	Room 1

Thursday 18th March 2004
14:30 to 15:30	Bob Vaughan Mean value theorems for primes in arithmetic progressions	Room 2
16:00 to 17:00	Yoichi Motohashi Nihon University Feasibility of a unified treatment of mean values of automorphic L-functions	Room 2

Wednesday 24th March 2004
16:00 to 17:00	Oriol Bohigas Spectral properties of distance matrices	Room 2

Thursday 25th March 2004
16:00 to 17:00	Akio Fujii Rikkyo University Some problems on the distribution of the zeros of the Riemann zeta function	Room 1

Thursday 15th April 2004
14:30 to 15:30	The partition function p(n) and the many-body density of states	Room 1
16:00 to 17:00	Peter Forrester Applications and generalisations of Fisher-Hartwig asymptotics	Room 1

Thursday 29th April 2004
14:30 to 15:30	Martin Zirnbauer Universität zu Köln From random matrices to supermanifolds	Room 2
16:00 to 17:00	John Harnad Max integrals as isomonodromic tau functi	Room 2

Tuesday 11th May 2004
14:30 to 15:30	Kambiz Farahmand Random polynomials	Room 1

Thursday 13th May 2004
16:00 to 17:00	Boris Khoruzhenko Queen Mary University of London Random determinants and eigenvalue distributions in the complex plane	Room 1

Thursday 27th May 2004
14:30 to 15:30	Yuri Suhov University of Cambridge Anderson localisation for multi-particle lattice systems	Room 1
16:00 to 17:00	Ali Ozluk University of Maine Low-lying zeroes of quadratic L-functions	Room 1

Thursday 3rd June 2004
14:30 to 15:30	Victor Perez-Abreu CIMAT (Centro de Investigación en Matemáticas), Mexico Type G ensembles of random matrices	Room 1
16:00 to 17:00	Eduardo Duenez Symmetry flipping in families of L-functions	Room 1

Thursday 10th June 2004
16:00 to 17:00	On the moments of traces of matrices of classical groups	Room 1

Thursday 17th June 2004
14:30 to 15:30	Alexander Soshnikov Poisson statistics for the largest eigenvalues in Wigner and sample covariance random matrices with heavy tails	Room 1
16:00 to 17:00	Triangulations of surfaces and distributing points on a sphere	Room 1

Tuesday 22nd June 2004
16:00 to 17:00	Alexander Its Indiana University Painleve transcendents and random matrices	Room 2

Thursday 24th June 2004
16:00 to 17:00	Dennis Hejhal Multi-variate Gaussians for L-functions and applications to zeros	Room 1

Thursday 8th July 2004
14:30 to 15:30	Andrew Granville Université de Montréal Uncertainty principles in arithmetic	Room 1

Random matrix approaches in number theory

Organisers

Brian Conrey
Persi Diaconis Stanford University
Francesco Mezzadri University of Bristol
Peter Sarnak Princeton University
Nina Snaith University of Bristol

Speakers

Oriol Bohigas
Eduardo Duenez
Kambiz Farahmand
David Farmer
Peter Forrester
Akio Fujii Rikkyo University
Alex Gamburd Stanford University
Steve Gonek University of Rochester
Andrew Granville University of Georgia
John Harnad Université de Montréal
Dennis Hejhal
Christopher Hughes
Alexander Its Indiana University
Jonathan Keating University of Bristol
Boris Khoruzhenko Queen Mary University of London
Shin-ya Koyama Keio University
Yoichi Motohashi Nihon University
Ali Ozluk University of Maine
Victor Perez-Abreu CIMAT (Centro de Investigación en Matemáticas), Mexico
Alexander Soshnikov University of California
Yuri Suhov University of Cambridge
Bob Vaughan
Mark Watkins Pennsylvania State University
Martin Zirnbauer Universität zu Köln

Show All

Random matrix approaches in number theory

Programme theme

Preprints

Random matrix theory and entanglement in quantum spin chains

On the moments of traces of matrices of classical groups

The zeros of random polynomials cluster uniformly near the unit circle

Mock-Gaussian behaviour

Symmetry classes of disordered fermions

Explicit lower bounds on the modular degree of an elliptic curve

Workshops

Ranks of elliptic curves and random matrix theory

Recent perspectives in random matrix theory and number theory

Random matrices and probability

Random matrix theory and arithmetic aspects of quantum chaos

Matrix ensembles and L-functions

Seminars

Organisers

Speakers

Supported By

Random matrix approaches in number theory

Programme theme

Preprints

Random matrix theory and entanglement in quantum spin chains

On the moments of traces of matrices of classical groups

The zeros of random polynomials cluster uniformly near the unit circle

Mock-Gaussian behaviour

Symmetry classes of disordered fermions

Explicit lower bounds on the modular degree of an elliptic curve

Workshops

Ranks of elliptic curves and random matrix theory

Recent perspectives in random matrix theory and number theory

Random matrices and probability

Random matrix theory and arithmetic aspects of quantum chaos

Matrix ensembles and L-functions

Seminars

Organisers

Speakers

Supported By

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