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Seminars (RMAW02)

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Event When Speaker Title Presentation Material
RMAW02 29th March 2004
10:00 to 11:00
Prime number theory & the Riemann zeta-function 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 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

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 zeta-function 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 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

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 L-functions
RMAW02 30th March 2004
09:00 to 10:00
Prime number theory & the Riemann zeta-function 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 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

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 zeta-function 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 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

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 zeta-function 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 zeta-function 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 n-point 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 zeta-function 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 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

RMAW02 1st April 2004
10:00 to 11:00
Pair correlation of zeros of the Riemann zeta-function 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 n-point correlation function for the Riemann zeros II
RMAW02 1st April 2004
15:30 to 16:30
Pair correlation of zeros of the Riemann zeta-function 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
L-functions 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 zeta-function 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 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

RMAW02 2nd April 2004
11:30 to 12:30
Heuristic derivation of the n-point 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 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.

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 L-function central values and RMT predictions

RMAW02 5th April 2004
10:00 to 11:00
B Conrey Statistics of low-lying zeros of L-function 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 L-function 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 L-functions 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 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.

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 low-lying zeros of L-function and random matrix theory II
RMAW02 6th April 2004
14:00 to 15:00
M Rubinstein Computational methods for L-functions II
RMAW02 6th April 2004
15:30 to 16:30
B Conrey Statistics of low-lying zeros of L-function 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 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.

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 low-lying zeros of L-function and random matrix theory IV
RMAW02 7th April 2004
11:30 to 12:30
M Rubinstein Computational methods for L-functions 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 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.

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 Mock-Gaussian behaviour

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

RMAW02 8th April 2004
10:00 to 11:00
Families \& conjectures for moments of L-functions
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 L-function central values and RMT predictions

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