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Random Matrix Approaches in Number Theory

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

26th January 2004 to 16th July 2004
Nina Snaith University of Bristol
Brian Conrey [Palo Alto], [American Institute of Mathematics]
Persi Diaconis Stanford University
Francesco Mezzadri University of Bristol
Peter Sarnak Princeton University


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.

Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons