Videos and presentation materials from other INI events are also available.
Event  When  Speaker  Title  Presentation Material 

RMAW02 
29th March 2004 10:00 to 11:00 
Prime number theory & the Riemann zetafunction I 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 

RMAW02 
29th March 2004 11:30 to 12:30 
Y Fyodorov  Gaussian ensembles of random matrices I  
RMAW02 
29th March 2004 14:00 to 15:00 
Prime number theory \& the Riemann zetafunction II 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 

RMAW02 
29th March 2004 15:30 to 16:30 
Y Fyodorov  Gaussian ensembles of random matrices II  
RMAW02 
29th March 2004 16:30 to 17:30 
Artin Lfunctions  
RMAW02 
30th March 2004 09:00 to 10:00 
Prime number theory & the Riemann zetafunction III 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 

RMAW02 
30th March 2004 10:00 to 11:00 
Elliptic curves  
RMAW02 
30th March 2004 11:30 to 12:30 
Y Fyodorov  Gaussian ensembles of random matrices III  
RMAW02 
30th March 2004 14:30 to 15:30 
Prime number theory & the Riemann zetafunction IV 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 

RMAW02 
30th March 2004 16:00 to 17:00 
Y Fyodorov  Gaussian ensembles of random matrices IV  
RMAW02 
30th March 2004 17:00 to 18:00 
Pair correlation of zeros of the Riemann zetafunction and prime numbers I 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


RMAW02 
31st March 2004 09:00 to 10:00 
Pair correlation of zeros of the Riemann zetafunction and prime numbers II 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


RMAW02 
31st March 2004 10:00 to 11:00 
Heuristic derivation of the npoint correlation function for the Riemann zeros I  
RMAW02 
31st March 2004 11:30 to 12:30 
Modular forms  
RMAW02 
1st April 2004 09:00 to 10:00 
Prime number theory \& the Riemann zetafunction V 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 

RMAW02 
1st April 2004 10:00 to 11:00 
Pair correlation of zeros of the Riemann zetafunction and prime numbers III 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


RMAW02 
1st April 2004 11:30 to 12:30 
Y Fyodorov  Gaussian ensembles of random matrices V  
RMAW02 
1st April 2004 14:00 to 15:00 
Heuristic derivation of the npoint correlation function for the Riemann zeros II  
RMAW02 
1st April 2004 15:30 to 16:30 
Pair correlation of zeros of the Riemann zetafunction and prime numbers IV 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


RMAW02 
1st April 2004 16:30 to 17:30 
Lfunctions over functions fields  
RMAW02 
2nd April 2004 09:00 to 10:00 
Y Fyodorov  Gaussian ensembles of random matrices VI  
RMAW02 
2nd April 2004 10:00 to 11:00 
Prime number theory \& the Riemann zetafunction VI 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 

RMAW02 
2nd April 2004 11:30 to 12:30 
Heuristic derivation of the npoint correlation function for the Riemann zeros III  
RMAW02 
2nd April 2004 14:00 to 15:00 
OG Bohigas 
Compund nucleus resonances, random matrices, quantum chaos 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. 

RMAW02 
2nd April 2004 15:30 to 16:30 
Quantum chaology and zeta 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. 

RMAW02 
3rd April 2004 10:00 to 11:00 
Mean values & zeros of the zeta function  
RMAW02 
3rd April 2004 11:30 to 12:30 
Toeplitz determinants & connections to random matrices I 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. 

RMAW02 
3rd April 2004 13:30 to 14:30 
Spacing distributions for random matrix ensembles I 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


RMAW02 
5th April 2004 09:00 to 10:00 
RMT moment calculations I 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 

RMAW02 
5th April 2004 10:00 to 11:00 
B Conrey  Statistics of lowlying zeros of Lfunction and random matrix theory I  
RMAW02 
5th April 2004 11:30 to 12:30 
Low moments of the Riemann zeta function  
RMAW02 
5th April 2004 14:00 to 15:00 
RMT moment calculations II 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 

RMAW02 
5th April 2004 15:30 to 16:30 
C Hughes 
Derivatives of the Riemann zeta function 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. 

RMAW02 
5th April 2004 16:30 to 17:30 
M Rubinstein  Computational methods for Lfunctions I  
RMAW02 
6th April 2004 09:00 to 10:00 
E Basor 
Toeplitz determinants \& connections to random matrices II 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. 

RMAW02 
6th April 2004 10:00 to 11:00 
Spacing distributions for random matrix ensembles II 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


RMAW02 
6th April 2004 11:30 to 12:30 
B Conrey  Statistics of lowlying zeros of Lfunction and random matrix theory II  
RMAW02 
6th April 2004 14:00 to 15:00 
M Rubinstein  Computational methods for Lfunctions II  
RMAW02 
6th April 2004 15:30 to 16:30 
B Conrey  Statistics of lowlying zeros of Lfunction and random matrix theory III  
RMAW02 
6th April 2004 16:30 to 17:30 
Toeplitz determinants \& connections to random matrices III 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. 

RMAW02 
7th April 2004 09:00 to 10:00 
Mean values of Dirichlet polynomials \& applications  
RMAW02 
7th April 2004 10:00 to 11:00 
B Conrey  Statistics of lowlying zeros of Lfunction and random matrix theory IV  
RMAW02 
7th April 2004 11:30 to 12:30 
M Rubinstein  Computational methods for Lfunctions III  
RMAW02 
7th April 2004 14:00 to 15:00 
C Hughes 
A new model for the Riemann zeta function 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. 

RMAW02 
7th April 2004 15:30 to 16:30 
Spacing distributions for random matrix ensembles III 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


RMAW02 
8th April 2004 09:00 to 10:00 
C Hughes 
MockGaussian behaviour 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


RMAW02 
8th April 2004 10:00 to 11:00 
Families \& conjectures for moments of Lfunctions  
RMAW02 
8th April 2004 11:30 to 12:30 
RMT moment calculations III 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 