RMA
26 January 2004 to 16 July 2004
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
Title | Year | Programme | |
---|---|---|---|
Random matrix theory and entanglement in quantum spin chainsAuthors: Francesco Mezzadri, Jon Keating |
2003 | RMA | 21 October 2016 |
On the moments of traces of matrices of classical groupsAuthors: V Vasilchuk, L Pastur |
2003 | RMA | 21 October 2016 |
The zeros of random polynomials cluster uniformly near the unit circleAuthors: Ashkan Nikeghbali, Christopher Hughes |
2003 | RMA | 21 October 2016 |
Mock-Gaussian behaviourAuthors: Christopher Hughes |
2003 | RMA | 21 October 2016 |
Symmetry classes of disordered fermionsAuthors: A Huckleberry, Martin Zirnbauer, P Heinzner |
2003 | RMA | 21 October 2016 |
Explicit lower bounds on the modular degree of an elliptic curveAuthors: Mark Watkins |
2003 | RMA | 21 October 2016 |
9 February 2004 to 13 February 2004
29 March 2004 to 8 April 2004
18 May 2004 to 21 May 2004
28 June 2004 to 2 July 2004
12 July 2004 to 16 July 2004
Thursday 29th January 2004 | |||
---|---|---|---|
15:00 to 16:00 | 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 |
Room 1 | |
16:30 to 17:30 |
Christopher Hughes |
Room 1 |
Thursday 19th February 2004 | |||
---|---|---|---|
14:30 to 15:30 |
Steve Gonek University of Rochester |
Room 1 | |
16:00 to 17:00 |
David Farmer |
Room 1 |
Thursday 26th February 2004 | |||
---|---|---|---|
16:00 to 17:00 |
Shin-ya Koyama Keio University |
Room 1 |
Thursday 4th March 2004 | |||
---|---|---|---|
14:30 to 15:30 |
Jonathan Keating University of Bristol |
Room 1 | |
16:00 to 17:00 |
Alex Gamburd Stanford University |
Room 1 |
Thursday 11th March 2004 | |||
---|---|---|---|
16:00 to 17:00 | Room 1 |
Thursday 18th March 2004 | |||
---|---|---|---|
14:30 to 15:30 |
Bob Vaughan |
Room 2 | |
16:00 to 17:00 |
Yoichi Motohashi Nihon University |
Room 2 |
Wednesday 24th March 2004 | |||
---|---|---|---|
16:00 to 17:00 |
Oriol Bohigas |
Room 2 |
Thursday 25th March 2004 | |||
---|---|---|---|
16:00 to 17:00 |
Akio Fujii Rikkyo University |
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 |
Room 1 |
Thursday 29th April 2004 | |||
---|---|---|---|
14:30 to 15:30 |
Martin Zirnbauer Universität zu Köln |
Room 2 | |
16:00 to 17:00 |
John Harnad |
Room 2 |
Tuesday 11th May 2004 | |||
---|---|---|---|
14:30 to 15:30 |
Kambiz Farahmand |
Room 1 |
Thursday 13th May 2004 | |||
---|---|---|---|
16:00 to 17:00 |
Boris Khoruzhenko Queen Mary University of London |
Room 1 |
Thursday 27th May 2004 | |||
---|---|---|---|
14:30 to 15:30 |
Yuri Suhov University of Cambridge |
Room 1 | |
16:00 to 17:00 |
Ali Ozluk University of Maine |
Room 1 |
Thursday 3rd June 2004 | |||
---|---|---|---|
14:30 to 15:30 |
Victor Perez-Abreu CIMAT (Centro de Investigación en Matemáticas), Mexico |
Room 1 | |
16:00 to 17:00 |
Eduardo Duenez |
Room 1 |
Thursday 10th June 2004 | |||
---|---|---|---|
16:00 to 17:00 | Room 1 |
Thursday 17th June 2004 | |||
---|---|---|---|
14:30 to 15:30 |
Alexander Soshnikov |
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 |
Room 2 |
Thursday 24th June 2004 | |||
---|---|---|---|
16:00 to 17:00 |
Dennis Hejhal |
Room 1 |
Thursday 8th July 2004 | |||
---|---|---|---|
14:30 to 15:30 |
Andrew Granville Université de Montréal |
Room 1 |
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