Recent Perspectives in Random Matrix Theory and Number Theory
Monday 29th March 2004 to Thursday 8th April 2004
| 10:00 to 11:00 |
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 |
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| 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 |
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| 14:00 to 15:00 |
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 |
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| 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 |
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| 16:30 to 17:30 |
Artin L-functions Session: Recent Perspectives in Random Matrix Theory and Number Theory |
INI 1 |
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| 09:00 to 10:00 |
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 |
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| 10:00 to 11:00 |
Elliptic curves Session: Recent Perspectives in Random Matrix Theory and Number Theory |
INI 1 |
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| 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 |
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| 14:30 to 15:30 |
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 |
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| 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 |
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| 17:00 to 18:00 |
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
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INI 1 |
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| 09:00 to 10:00 |
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
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INI 1 |
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| 10:00 to 11:00 |
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 |
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| 11:30 to 12:30 |
Modular forms Session: Recent Perspectives in Random Matrix Theory and Number Theory |
INI 1 |
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| 09:00 to 10:00 |
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 |
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| 10:00 to 11:00 |
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
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INI 1 |
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| 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 |
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| 14:00 to 15:00 |
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 |
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| 15:30 to 16:30 |
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
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INI 1 |
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| 16:30 to 17:30 |
L-functions over functions fields Session: Recent Perspectives in Random Matrix Theory and Number Theory |
INI 1 |
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| 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 |
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| 10:00 to 11:00 |
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 |
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| 11:30 to 12:30 |
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 |
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| 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 |
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| 15:30 to 16:30 |
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 |
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| 10:00 to 11:00 |
Mean values & zeros of the zeta function Session: Recent Perspectives in Random Matrix Theory and Number Theory |
INI 1 |
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| 11:30 to 12:30 |
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 |
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| 13:30 to 14:30 |
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
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INI 1 |
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| 09:00 to 10:00 |
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 |
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| 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 |
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| 11:30 to 12:30 |
Low moments of the Riemann zeta function Session: Recent Perspectives in Random Matrix Theory and Number Theory |
INI 1 |
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| 14:00 to 15:00 |
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 |
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| 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 |
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| 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 |
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| 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 |
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| 10:00 to 11:00 |
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
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INI 1 |
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| 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 |
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| 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 |
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| 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 |
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| 16:30 to 17:30 |
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 |
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| 09:00 to 10:00 |
Mean values of Dirichlet polynomials \& applications Session: Recent Perspectives in Random Matrix Theory and Number Theory |
INI 1 |
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| 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 |
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| 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 |
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| 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 |
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| 15:30 to 16:30 |
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
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INI 1 |
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| 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
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INI 1 |
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| 10:00 to 11:00 |
Families \& conjectures for moments of L-functions Session: Recent Perspectives in Random Matrix Theory and Number Theory |
INI 1 |
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| 11:30 to 12:30 |
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 |
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