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Timetable (RMAW02)

Recent Perspectives in Random Matrix Theory and Number Theory

Monday 29th March 2004 to Thursday 8th April 2004

Monday 29th March 2004
10:00 to 11:00 R Heath-Brown ([Oxford])
Prime number theory & the Riemann zeta-function I
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture 1:-

Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model

Lecture 2:-

Open questions on primes Recent achievements of prime number theory The Riemann Zeta-function Euler product Analytic continuation and functional equation (via theta function)

Lecture 3:-

Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T)

Lecture 4:-

N(T) and S(T) (continued) Non-vanishing on the 1-line Proof of PNT

Lecture 5:-

Proof of PNT (continued) Weil type Explicit formulae

Lecture 6:-

Characters Dirichlet L-functions

Pre-requisites:-

Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD)

Recommended text:-

Davenport, Multiplicative Number Theory

INI 1
11:30 to 12:30 Y Fyodorov ([Brunel])
Gaussian ensembles of random matrices I
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
14:00 to 15:00 R Heath-Brown ([Oxford])
Prime number theory \& the Riemann zeta-function II
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture 1:-

Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model

Lecture 2:-

Open questions on primes Recent achievements of prime number theory The Riemann Zeta-function Euler product Analytic continuation and functional equation (via theta function)

Lecture 3:-

Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T)

Lecture 4:-

N(T) and S(T) (continued) Non-vanishing on the 1-line Proof of PNT

Lecture 5:-

Proof of PNT (continued) Weil type Explicit formulae

Lecture 6:-

Characters Dirichlet L-functions

Pre-requisites:-

Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD)

Recommended text:-

Davenport, Multiplicative Number Theory

INI 1
15:30 to 16:30 Y Fyodorov ([Brunel])
Gaussian ensembles of random matrices II
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
16:30 to 17:30 P Michel ([Montpellier II])
Artin L-functions
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
Tuesday 30th March 2004
09:00 to 10:00 R Heath-Brown ([Oxford])
Prime number theory & the Riemann zeta-function III
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture 1:-

Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model

Lecture 2:-

Open questions on primes Recent achievements of prime number theory The Riemann Zeta-function Euler product Analytic continuation and functional equation (via theta function)

Lecture 3:-

Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T)

Lecture 4:-

N(T) and S(T) (continued) Non-vanishing on the 1-line Proof of PNT

Lecture 5:-

Proof of PNT (continued) Weil type Explicit formulae

Lecture 6:-

Characters Dirichlet L-functions

Pre-requisites:-

Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD)

Recommended text:-

Davenport, Multiplicative Number Theory

INI 1
10:00 to 11:00 P Michel ([Montpellier II])
Elliptic curves
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
11:30 to 12:30 Y Fyodorov ([Brunel])
Gaussian ensembles of random matrices III
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
14:30 to 15:30 R Heath-Brown ([Oxford])
Prime number theory & the Riemann zeta-function IV
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture 1:-

Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model

Lecture 2:-

Open questions on primes Recent achievements of prime number theory The Riemann Zeta-function Euler product Analytic continuation and functional equation (via theta function)

Lecture 3:-

Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T)

Lecture 4:-

N(T) and S(T) (continued) Non-vanishing on the 1-line Proof of PNT

Lecture 5:-

Proof of PNT (continued) Weil type Explicit formulae

Lecture 6:-

Characters Dirichlet L-functions

Pre-requisites:-

Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD)

Recommended text:-

Davenport, Multiplicative Number Theory

INI 1
16:00 to 17:00 Y Fyodorov ([Brunel])
Gaussian ensembles of random matrices IV
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
17:00 to 18:00 DA Goldston ([San Jose State])
Pair correlation of zeros of the Riemann zeta-function and prime numbers I
Session: Recent Perspectives in Random Matrix Theory and Number Theory

The first lecture will cover Montgomery's 1973 paper where the pair correlation and GUE conjecture were first made, together with some applications. The second and third lecture will cover the connections between zeros and prime numbers, and include further methods for examining the distribution of primes. The fourth lecture will give a brief introduction to Selberg's theory of the distribution of values of log (zeta) and S(t).

Related Links

INI 1
Wednesday 31st March 2004
09:00 to 10:00 DA Goldston ([San Jose State])
Pair correlation of zeros of the Riemann zeta-function and prime numbers II
Session: Recent Perspectives in Random Matrix Theory and Number Theory

The first lecture will cover Montgomery's 1973 paper where the pair correlation and GUE conjecture were first made, together with some applications. The second and third lecture will cover the connections between zeros and prime numbers, and include further methods for examining the distribution of primes. The fourth lecture will give a brief introduction to Selberg's theory of the distribution of values of log (zeta) and S(t).

Related Links

INI 1
10:00 to 11:00 EB Bogomolny ([Paris Sud])
Heuristic derivation of the n-point correlation function for the Riemann zeros I
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
11:30 to 12:30 P Michel ([Montpellier II])
Modular forms
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
Thursday 1st April 2004
09:00 to 10:00 R Heath-Brown ([Oxford])
Prime number theory \& the Riemann zeta-function V
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture 1:-

Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model

Lecture 2:-

Open questions on primes Recent achievements of prime number theory The Riemann Zeta-function Euler product Analytic continuation and functional equation (via theta function)

Lecture 3:-

Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T)

Lecture 4:-

N(T) and S(T) (continued) Non-vanishing on the 1-line Proof of PNT

Lecture 5:-

Proof of PNT (continued) Weil type Explicit formulae

Lecture 6:-

Characters Dirichlet L-functions

Pre-requisites:-

Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD)

Recommended text:-

Davenport, Multiplicative Number Theory

INI 1
10:00 to 11:00 DA Goldston ([San Jose State])
Pair correlation of zeros of the Riemann zeta-function and prime numbers III
Session: Recent Perspectives in Random Matrix Theory and Number Theory

The first lecture will cover Montgomery's 1973 paper where the pair correlation and GUE conjecture were first made, together with some applications. The second and third lecture will cover the connections between zeros and prime numbers, and include further methods for examining the distribution of primes. The fourth lecture will give a brief introduction to Selberg's theory of the distribution of values of log (zeta) and S(t).

Related Links

INI 1
11:30 to 12:30 Y Fyodorov ([Brunel])
Gaussian ensembles of random matrices V
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
14:00 to 15:00 EB Bogomolny ([Paris Sud])
Heuristic derivation of the n-point correlation function for the Riemann zeros II
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
15:30 to 16:30 DA Goldston ([San Jose State])
Pair correlation of zeros of the Riemann zeta-function and prime numbers IV
Session: Recent Perspectives in Random Matrix Theory and Number Theory

The first lecture will cover Montgomery's 1973 paper where the pair correlation and GUE conjecture were first made, together with some applications. The second and third lecture will cover the connections between zeros and prime numbers, and include further methods for examining the distribution of primes. The fourth lecture will give a brief introduction to Selberg's theory of the distribution of values of log (zeta) and S(t).

Related Links

INI 1
16:30 to 17:30 P Michel ([Montpellier II])
L-functions over functions fields
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
Friday 2nd April 2004
09:00 to 10:00 Y Fyodorov ([Brunel])
Gaussian ensembles of random matrices VI
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
10:00 to 11:00 R Heath-Brown ([Oxford])
Prime number theory \& the Riemann zeta-function VI
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture 1:-

Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model

Lecture 2:-

Open questions on primes Recent achievements of prime number theory The Riemann Zeta-function Euler product Analytic continuation and functional equation (via theta function)

Lecture 3:-

Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T)

Lecture 4:-

N(T) and S(T) (continued) Non-vanishing on the 1-line Proof of PNT

Lecture 5:-

Proof of PNT (continued) Weil type Explicit formulae

Lecture 6:-

Characters Dirichlet L-functions

Pre-requisites:-

Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD)

Recommended text:-

Davenport, Multiplicative Number Theory

INI 1
11:30 to 12:30 EB Bogomolny ([Paris Sud])
Heuristic derivation of the n-point correlation function for the Riemann zeros III
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
14:00 to 15:00 OG Bohigas ([Paris Sud])
Compund nucleus resonances, random matrices, quantum chaos
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Wigner introduced random matrices in physics when searching for a guiding principle to understand properties of the compound nucleus resonances. At the end the experimental observations turned out to be remarkably consistent with random matrix theory predictions. Could random matrix theory be justified in dynamical terms? To answer this question deep connections between quantum bahaviour of classically chaotic systems (quantum chaos) and random matrices have been established. Open problems still remain. Some highlights of this long excursion, covering more than fourty years, will be given.

INI 1
15:30 to 16:30 MV Berry ([Bristol])
Quantum chaology and zeta
Session: Recent Perspectives in Random Matrix Theory and Number Theory

As requested by the organisers the talk will consist of my reminiscences of how the different intellectual streams of quantum chaology and Riemannology became intermingled, with benefit to both.

INI 1
Saturday 3rd April 2004
10:00 to 11:00 S Gonek ([Rochester])
Mean values & zeros of the zeta function
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
11:30 to 12:30 E Basor ([California Polytechnic State])
Toeplitz determinants & connections to random matrices I
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture1: This lecture will contain an introduction of Toeplitz Operators, Hankel Operators, and finite Toeplitz matrices. We will give a simple proof of the Strong Szego Limit Theorem which describes an asymptotic formula for determinants of Toeplitz matrices with smooth symbols.

Lecture2: This lecture will show how to extend the Szego Limit Theorem to symbols with discontinuities, the class of Fisher-Hartwig symbols.

Lecture3: The final lecture will show the connection of the limit theorems to results about the distribution functions for random matrix ensembles. If time permits, Laguerre ensembles will be considered and also the analogues for Wiener-Hopf opeators.

INI 1
13:30 to 14:30 P Forrester ([Melbourne])
Spacing distributions for random matrix ensembles I
Session: Recent Perspectives in Random Matrix Theory and Number Theory

The calculation of eigenvalue spacing distributions for classical random matrix ensembles with unitary symmetry is intimately related to the theory of integrable systems and Painleve' equations. These theories provide the characterization of spacing distributions as solutions of nonlinear equations solvable in terms of Painleve' transcendents. Two approaches to achieve this goal will be detailed. One approach makes use of function theoretic properties of Fredholm determinants. The other proceeds via the theory of Painleve' systems.

Related Links

INI 1
Monday 5th April 2004
09:00 to 10:00 JP Keating ([Bristol])
RMT moment calculations I
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture 1: Introduction to the CUE and averages over other classical compact groups

Lecture 2: moments and value distribution of CUE characteristic polynomials; comparison with the moments and value distribution of the Riemann zeta function

Lecture 3: vanishing of L-function central values and RMT predictions

INI 1
10:00 to 11:00 B Conrey ([AIM])
Statistics of low-lying zeros of L-function and random matrix theory I
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
11:30 to 12:30 DW Farmer ([AIM])
Low moments of the Riemann zeta function
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
14:00 to 15:00 JP Keating ([Bristol])
RMT moment calculations II
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture 1: Introduction to the CUE and averages over other classical compact groups

Lecture 2: moments and value distribution of CUE characteristic polynomials; comparison with the moments and value distribution of the Riemann zeta function

Lecture 3: vanishing of L-function central values and RMT predictions

INI 1
15:30 to 16:30 C Hughes ([AIM])
Derivatives of the Riemann zeta function
Session: Recent Perspectives in Random Matrix Theory and Number Theory

In this lecture we will look at moments of the derivative of the Riemann zeta function, and how random matrix theory can help predict their asymptotic behaviour. Applications to estimating the size of large gaps between the zeros will also be discussed.

INI 1
16:30 to 17:30 M Rubinstein ([Waterloo, Canada])
Computational methods for L-functions I
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
Tuesday 6th April 2004
09:00 to 10:00 E Basor ([California Poltechnic State])
Toeplitz determinants \& connections to random matrices II
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture1: This lecture will contain an introduction of Toeplitz Operators, Hankel Operators, and finite Toeplitz matrices. We will give a simple proof of the Strong Szego Limit Theorem which describes an asymptotic formula for determinants of Toeplitz matrices with smooth symbols.

Lecture2: This lecture will show how to extend the Szego Limit Theorem to symbols with discontinuities, the class of Fisher-Hartwig symbols.

Lecture3: The final lecture will show the connection of the limit theorems to results about the distribution functions for random matrix ensembles. If time permits, Laguerre ensembles will be considered and also the analogues for Wiener-Hopf opeators.

INI 1
10:00 to 11:00 P Forrester ([Melbourne])
Spacing distributions for random matrix ensembles II
Session: Recent Perspectives in Random Matrix Theory and Number Theory

The calculation of eigenvalue spacing distributions for classical random matrix ensembles with unitary symmetry is intimately related to the theory of integrable systems and Painleve' equations. These theories provide the characterization of spacing distributions as solutions of nonlinear equations solvable in terms of Painleve' transcendents. Two approaches to achieve this goal will be detailed. One approach makes use of function theoretic properties of Fredholm determinants. The other proceeds via the theory of Painleve' systems.

Related Links

INI 1
11:30 to 12:30 B Conrey ([AIM])
Statistics of low-lying zeros of L-function and random matrix theory II
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
14:00 to 15:00 M Rubinstein ([Waterloo, Canada])
Computational methods for L-functions II
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
15:30 to 16:30 B Conrey ([AIM])
Statistics of low-lying zeros of L-function and random matrix theory III
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
16:30 to 17:30 E Basor ([California Polytechnic State])
Toeplitz determinants \& connections to random matrices III
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture1: This lecture will contain an introduction of Toeplitz Operators, Hankel Operators, and finite Toeplitz matrices. We will give a simple proof of the Strong Szego Limit Theorem which describes an asymptotic formula for determinants of Toeplitz matrices with smooth symbols.

Lecture2: This lecture will show how to extend the Szego Limit Theorem to symbols with discontinuities, the class of Fisher-Hartwig symbols.

Lecture3: The final lecture will show the connection of the limit theorems to results about the distribution functions for random matrix ensembles. If time permits, Laguerre ensembles will be considered and also the analogues for Wiener-Hopf opeators.

INI 1
Wednesday 7th April 2004
09:00 to 10:00 S Gonek ([Rochester])
Mean values of Dirichlet polynomials \& applications
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
10:00 to 11:00 B Conrey ([AIM])
Statistics of low-lying zeros of L-function and random matrix theory IV
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
11:30 to 12:30 M Rubinstein ([Waterloo, Canada])
Computational methods for L-functions III
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
14:00 to 15:00 C Hughes ([AIM])
A new model for the Riemann zeta function
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Random matrix theory (RMT) has been very successul at modeling the zeros of the zeta function. A recent conjecture of Keating and Snaith uses RMT to conjecture the asymptotic form of moments of the Riemann zeta function, but the conjecture requires an ad-hoc addition from primes to fit known results. In this lecture a new model for the zeta function will be presented, where it is writen as a partial Euler product times a partial Hadamard product. This model enables us to rederive the Keating-Snaith conjecture with both the prime contribution and the random matrix contribution appearing naturally. The research presented in this lecture is joint with Jon Keating and Steve Gonek.

INI 1
15:30 to 16:30 P Forrester ([Melbourne])
Spacing distributions for random matrix ensembles III
Session: Recent Perspectives in Random Matrix Theory and Number Theory

The calculation of eigenvalue spacing distributions for classical random matrix ensembles with unitary symmetry is intimately related to the theory of integrable systems and Painleve' equations. These theories provide the characterization of spacing distributions as solutions of nonlinear equations solvable in terms of Painleve' transcendents. Two approaches to achieve this goal will be detailed. One approach makes use of function theoretic properties of Fredholm determinants. The other proceeds via the theory of Painleve' systems.

Related Links

INI 1
Thursday 8th April 2004
09:00 to 10:00 C Hughes ([AIM])
Mock-Gaussian behaviour
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Mock-Gaussian behaviour is when a smooth counting function (or linear statistic) has its first few moments equal to the moments of a Gaussian distribution, even though it is not a normal distribution. In this lecture we will see that this behaviour holds eigenvalues of random matrices, and analogously for the zeros of the Riemann zeta function and other L-functions. The research presented in this lecture is joint with Zeev Rudnick.

Related Links

INI 1
10:00 to 11:00 DW Farmer ([AIM])
Families \& conjectures for moments of L-functions
Session: Recent Perspectives in Random Matrix Theory and Number Theory
INI 1
11:30 to 12:30 JP Keating ([Bristol])
RMT moment calculations III
Session: Recent Perspectives in Random Matrix Theory and Number Theory

Lecture 1: Introduction to the CUE and averages over other classical compact groups

Lecture 2: moments and value distribution of CUE characteristic polynomials; comparison with the moments and value distribution of the Riemann zeta function

Lecture 3: vanishing of L-function central values and RMT predictions

INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons