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Recent Perspectives in Random Matrix Theory and Number Theory

29th March 2004 to 8th April 2004

Organisers: Francesco Mezzadri (University of Bristol) Nina Snaith (University of Bristol)

A school run by The European Commission Research Training Network - Mathematical Aspects of Quantum Chaos

Purpose of the school

The connection between random matrix theory and the zeros of the Riemann zeta function was first suggested by Montgomery and Dyson in 1973, and later used in the 1980s to elucidate periodic orbit calculations in the field of quantum chaos. Just in the past few years it has also been employed to suggest brand new ways for predicting the behaviour of the Riemann zeta function and other L-functions. Notwithstanding these successes there has always been the problem that very few researchers are well-versed both in number theory and methods in mathematical physics. The aim of this school is to provide a grounding in both the relevant aspects of number theory, and the techniques of random matrix theory, as well as to inform the students of what progress has been made when these two apparently disparate subjects meet.

Topics to be covered

  • ensembles of Hermitian matrices
  • random matrix eigenvalue statistics
  • orthogonal polynomials
  • Painlevé theory and random matrix averages
  • Fisher-Hartwig conjecture and Toeplitz determinants
  • random matrix characteristic polynomials as models for L-functions
  • universality of random matrix ensembles
  • elementary number theory
  • the Riemann zeta function and Dirichlet L-functions
  • elliptic curves, automorphic forms and associated L-functions
  • computation of L-functions
  • Montgomery's conjecture on the two-point statistics of the Riemann zeta function
  • Selberg's theorem on the distribution of values of the logarithm of the Riemann zeta function
  • calculating moments of L-functions
  • recent work utilising random matrix theory in number theory
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Co-sponsored by:   Mathematical and Theoretical Physics group of the IOP and the National Science Foundation

 

University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons