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Workshop Programme

for period 29 March - 8 April 2004

Recent Perspectives in Random Matrix Theory and Number Theory

29 March - 8 April 2004

Timetable

Monday 29 March
Session: Recent Perspectives in Random Matrix Theory and Number Theory
10:00-11:00 Heath-Brown, R (Oxford)
  Prime number theory & the Riemann zeta-function I Sem 1
 

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

 
11:30-12:30 Fyodorov, Y (Brunel)
  Gaussian ensembles of random matrices I Sem 1
14:00-15:00 Heath-Brown, R (Oxford)
  Prime number theory \& the Riemann zeta-function II Sem 1
 

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

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

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

 
10:00-11:00 Michel, P (Montpellier II)
  Elliptic curves Sem 1
11:30-12:30 Fyodorov, Y (Brunel)
  Gaussian ensembles of random matrices III Sem 1
14:30-15:30 Heath-Brown, R (Oxford)
  Prime number theory & the Riemann zeta-function IV Sem 1
 

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

 
16:00-17:00 Fyodorov, Y (Brunel)
  Gaussian ensembles of random matrices IV Sem 1
17:00-18:00 Goldston, DA (San Jose State)
  Pair correlation of zeros of the Riemann zeta-function and prime numbers I Sem 1
 

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

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

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

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

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

 
10:00-11:00 Goldston, DA (San Jose State)
  Pair correlation of zeros of the Riemann zeta-function and prime numbers III Sem 1
 

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

 
11:30-12:30 Fyodorov, Y (Brunel)
  Gaussian ensembles of random matrices V Sem 1
14:00-15:00 Bogomolny, EB (Paris Sud)
  Heuristic derivation of the n-point correlation function for the Riemann zeros II Sem 1
15:30-16:30 Goldston, DA (San Jose State)
  Pair correlation of zeros of the Riemann zeta-function and prime numbers IV Sem 1
 

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

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

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

 
11:30-12:30 Bogomolny, EB (Paris Sud)
  Heuristic derivation of the n-point correlation function for the Riemann zeros III Sem 1
14:00-15:00 Bohigas, OG (Paris Sud)
  Compund nucleus resonances, random matrices, quantum chaos Sem 1
 

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.

 
15:30-16:30 Berry, MV (Bristol)
  Quantum chaology and zeta Sem 1
 

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.

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

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.

 
13:30-14:30 Forrester, P (Melbourne)
  Spacing distributions for random matrix ensembles I Sem 1
 

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

 
Monday 5 April
Session: Recent Perspectives in Random Matrix Theory and Number Theory
09:00-10:00 Keating, JP (Bristol)
  RMT moment calculations I Sem 1
 

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

 
10:00-11:00 Conrey, B (AIM)
  Statistics of low-lying zeros of L-function and random matrix theory I Sem 1
11:30-12:30 Farmer, DW (AIM)
  Low moments of the Riemann zeta function Sem 1
14:00-15:00 Keating, JP (Bristol)
  RMT moment calculations II Sem 1
 

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

 
15:30-16:30 Hughes, C (AIM)
  Derivatives of the Riemann zeta function Sem 1
 

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.

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

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.

 
10:00-11:00 Forrester, P (Melbourne)
  Spacing distributions for random matrix ensembles II Sem 1
 

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

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

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.

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

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.

 
15:30-16:30 Forrester, P (Melbourne)
  Spacing distributions for random matrix ensembles III Sem 1
 

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

 
Thursday 8 April
Session: Recent Perspectives in Random Matrix Theory and Number Theory
09:00-10:00 Hughes, C (AIM)
  Mock-Gaussian behaviour Sem 1
 

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

 
10:00-11:00 Farmer, DW (AIM)
  Families \& conjectures for moments of L-functions Sem 1
11:30-12:30 Keating, JP (Bristol)
  RMT moment calculations III Sem 1
 

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

 

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