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  
10:0011:00  HeathBrown, R (Oxford)  
Prime number theory & the Riemann zetafunction 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 Zetafunction 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) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

11:3012:30  Fyodorov, Y (Brunel)  
Gaussian ensembles of random matrices I  Sem 1  
14:0015:00  HeathBrown, R (Oxford)  
Prime number theory \& the Riemann zetafunction 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 Zetafunction 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) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

15:3016:30  Fyodorov, Y (Brunel)  
Gaussian ensembles of random matrices II  Sem 1  
16:3017:30  Michel, P (Montpellier II)  
Artin Lfunctions  Sem 1  
Tuesday 30 March  
Session: Recent Perspectives in Random Matrix Theory and Number Theory  
09:0010:00  HeathBrown, R (Oxford)  
Prime number theory & the Riemann zetafunction 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 Zetafunction 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) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

10:0011:00  Michel, P (Montpellier II)  
Elliptic curves  Sem 1  
11:3012:30  Fyodorov, Y (Brunel)  
Gaussian ensembles of random matrices III  Sem 1  
14:3015:30  HeathBrown, R (Oxford)  
Prime number theory & the Riemann zetafunction 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 Zetafunction 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) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

16:0017:00  Fyodorov, Y (Brunel)  
Gaussian ensembles of random matrices IV  Sem 1  
17:0018:00  Goldston, DA (San Jose State)  
Pair correlation of zeros of the Riemann zetafunction 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  
09:0010:00  Goldston, DA (San Jose State)  
Pair correlation of zeros of the Riemann zetafunction 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


10:0011:00  Bogomolny, EB (Paris Sud)  
Heuristic derivation of the npoint correlation function for the Riemann zeros I  Sem 1  
11:3012:30  Michel, P (Montpellier II)  
Modular forms  Sem 1  
Thursday 1 April  
Session: Recent Perspectives in Random Matrix Theory and Number Theory  
09:0010:00  HeathBrown, R (Oxford)  
Prime number theory \& the Riemann zetafunction 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 Zetafunction 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) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

10:0011:00  Goldston, DA (San Jose State)  
Pair correlation of zeros of the Riemann zetafunction 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


11:3012:30  Fyodorov, Y (Brunel)  
Gaussian ensembles of random matrices V  Sem 1  
14:0015:00  Bogomolny, EB (Paris Sud)  
Heuristic derivation of the npoint correlation function for the Riemann zeros II  Sem 1  
15:3016:30  Goldston, DA (San Jose State)  
Pair correlation of zeros of the Riemann zetafunction 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


16:3017:30  Michel, P (Montpellier II)  
Lfunctions over functions fields  Sem 1  
Friday 2 April  
Session: Recent Perspectives in Random Matrix Theory and Number Theory  
09:0010:00  Fyodorov, Y (Brunel)  
Gaussian ensembles of random matrices VI  Sem 1  
10:0011:00  HeathBrown, R (Oxford)  
Prime number theory \& the Riemann zetafunction 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 Zetafunction 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) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

11:3012:30  Bogomolny, EB (Paris Sud)  
Heuristic derivation of the npoint correlation function for the Riemann zeros III  Sem 1  
14:0015:00  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. 

15:3016:30  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  
10:0011:00  Gonek, S (Rochester)  
Mean values & zeros of the zeta function  Sem 1  
11:3012:30  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 FisherHartwig 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 WienerHopf opeators. 

13:3014:30  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  
09:0010:00  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 Lfunction central values and RMT predictions 

10:0011:00  Conrey, B (AIM)  
Statistics of lowlying zeros of Lfunction and random matrix theory I  Sem 1  
11:3012:30  Farmer, DW (AIM)  
Low moments of the Riemann zeta function  Sem 1  
14:0015:00  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 Lfunction central values and RMT predictions 

15:3016:30  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. 

16:3017:30  Rubinstein, M (Waterloo, Canada)  
Computational methods for Lfunctions I  Sem 1  
Tuesday 6 April  
Session: Recent Perspectives in Random Matrix Theory and Number Theory  
09:0010:00  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 FisherHartwig 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 WienerHopf opeators. 

10:0011:00  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


11:3012:30  Conrey, B (AIM)  
Statistics of lowlying zeros of Lfunction and random matrix theory II  Sem 1  
14:0015:00  Rubinstein, M (Waterloo, Canada)  
Computational methods for Lfunctions II  Sem 1  
15:3016:30  Conrey, B (AIM)  
Statistics of lowlying zeros of Lfunction and random matrix theory III  Sem 1  
16:3017:30  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 FisherHartwig 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 WienerHopf opeators. 

Wednesday 7 April  
Session: Recent Perspectives in Random Matrix Theory and Number Theory  
09:0010:00  Gonek, S (Rochester)  
Mean values of Dirichlet polynomials \& applications  Sem 1  
10:0011:00  Conrey, B (AIM)  
Statistics of lowlying zeros of Lfunction and random matrix theory IV  Sem 1  
11:3012:30  Rubinstein, M (Waterloo, Canada)  
Computational methods for Lfunctions III  Sem 1  
14:0015:00  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 adhoc 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 KeatingSnaith 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. 

15:3016:30  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  
09:0010:00  Hughes, C (AIM)  
MockGaussian behaviour  Sem 1  
MockGaussian 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 Lfunctions. The research presented in this lecture is joint with Zeev Rudnick. Related Links


10:0011:00  Farmer, DW (AIM)  
Families \& conjectures for moments of Lfunctions  Sem 1  
11:3012:30  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 Lfunction central values and RMT predictions 
