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Seminars (RMA)

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Event When Speaker Title Presentation Material
RMA 29th January 2004
15:00 to 16:00
M Stoltz Random matrices and invariant theory
RMA 29th January 2004
16:30 to 17:30
A Strombergsson Small solutions to linear congruences and Hecke equidistribution
RMA 5th February 2004
15:00 to 16:00
M Watkins Solving systems of polynomial equations via multidimensional p-adic Newton iteration
RMA 5th February 2004
16:30 to 17:30
C Hughes Moments of the Riemann Zeta function and random matrix theory "theory"
RMAW01 9th February 2004
11:00 to 11:45
The origins of the Birch/Swinnerton-Dyer conjecture: some personal reminiscences
RMAW01 9th February 2004
12:00 to 12:45
Ranks of elliptic curves
RMAW01 9th February 2004
14:30 to 15:15
C Delaunay Heuristics on Class groups and on Tate-Shafarevich groups
RMAW01 9th February 2004
15:30 to 16:15
M Rubinstein Moments, L-values & Ranks
RMAW01 9th February 2004
16:45 to 17:30
Vanishing of L-functions of elliptic curves over number fields
RMAW01 10th February 2004
09:30 to 18:00
Ranks of elliptic curves & random matrix theory
RMAW01 10th February 2004
11:15 to 18:00
Constructing rank 2 & rank 3 twists
RMAW01 11th February 2004
11:30 to 18:00
M Rubinstein Numerical evidence
RMAW01 13th February 2004
09:00 to 18:00
Heuristics for large rank
RMAW01 13th February 2004
10:00 to 18:00
M Watkins Numerical evidence
RMAW01 13th February 2004
14:30 to 18:00
C Hughes Using RMT to predict large values
RMA 19th February 2004
14:30 to 15:30
Mean value theorems and the zeros of the zeta function
RMA 19th February 2004
16:00 to 17:00
D Farmer Differentiation evens out zero spacings
RMA 26th February 2004
16:00 to 17:00
Multiple zeta functions
RMA 4th March 2004
14:30 to 15:30
Negative moments
RMA 4th March 2004
16:00 to 17:00
A Gamburd Random matrices, magic squares and the Riemann zeta function
RMA 11th March 2004
16:00 to 17:00
I Smolyarenko Parametric random matrix theory
RMA 18th March 2004
14:30 to 15:30
R Vaughan Mean value theorems for primes in arithmetic progressions
RMA 18th March 2004
16:00 to 17:00
Feasibility of a unified treatment of mean values of automorphic L-functions
RMA 24th March 2004
16:00 to 17:00
Spectral properties of distance matrices
RMA 25th March 2004
16:00 to 17:00
Some problems on the distribution of the zeros of the Riemann zeta function
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

RMA 15th April 2004
14:30 to 15:30
The partition function p(n) and the many-body density of states
RMA 15th April 2004
16:00 to 17:00
Applications and generalisations of Fisher-Hartwig asymptotics
RMA 29th April 2004
14:30 to 15:30
From random matrices to supermanifolds
RMA 29th April 2004
16:00 to 17:00
Max integrals as isomonodromic tau functi
RMA 11th May 2004
14:30 to 15:30
Random polynomials
RMA 13th May 2004
16:00 to 17:00
Random determinants and eigenvalue distributions in the complex plane
RMAW03 18th May 2004
09:00 to 10:00
Brownian motion in a Weyl chamber and GUE
RMAW03 18th May 2004
10:00 to 11:00
F Goetze Asymptotic spectral approximations
RMAW03 18th May 2004
10:30 to 11:00
B Khuruzhenko Moments of spectral determinants of random complex matrices
RMAW01 18th May 2004
13:00 to 14:00
H Widom Differencial equations for Dyson processes
RMAW01 18th May 2004
14:00 to 15:00
Granular Bosanization
RMAW03 19th May 2004
09:00 to 10:00
Integration over classical compact groups with applications to free probability and matrix integrals
RMAW03 19th May 2004
10:00 to 11:00
Central limit theorums for traces - a combinatorial and concentration approach
RMAW03 20th May 2004
09:00 to 10:00
Some properties of Wishart processes
RMAW03 20th May 2004
10:00 to 11:00
Some noncommutative central limit theorums
RMAW03 20th May 2004
11:00 to 12:00
Free transportation cost inequalities via random matrix approximation
RMAW03 20th May 2004
12:00 to 13:00
M Stolz Examples of dual pairs in random matrix theory
RMAW03 20th May 2004
13:00 to 14:00
Dyson's Brownian motion, interlacing and intertwining
RMAW03 21st May 2004
09:00 to 10:00
Y Doumerc Exit problems associated with finite reflection groups
RMAW01 21st May 2004
10:00 to 11:00
Y Fyodorov Complexity of random energy landscapes, glass transition and absolute value of spectral determinant of random matrices
RMAW03 21st May 2004
11:00 to 12:00
T Imamura Fluctuations of the 1D polynuclear growth model with external sources
RMAW03 21st May 2004
12:00 to 13:00
Universality of distributions from random matrix theory
RMAW03 21st May 2004
13:00 to 14:00
Asymptotics of Haar unitaries and their truncation
RMA 27th May 2004
14:30 to 15:30
Anderson localisation for multi-particle lattice systems
RMA 27th May 2004
16:00 to 17:00
Low-lying zeroes of quadratic L-functions
RMA 3rd June 2004
14:30 to 15:30
Type G ensembles of random matrices
RMA 3rd June 2004
16:00 to 17:00
Symmetry flipping in families of L-functions
RMA 10th June 2004
16:00 to 17:00
On the moments of traces of matrices of classical groups
RMA 17th June 2004
14:30 to 15:30
A Soshnikov Poisson statistics for the largest eigenvalues in Wigner and sample covariance random matrices with heavy tails
RMA 17th June 2004
16:00 to 17:00
Triangulations of surfaces and distributing points on a sphere
RMA 22nd June 2004
16:00 to 17:00
A Its Painleve transcendents and random matrices
RMA 24th June 2004
16:00 to 17:00
D Hejhal Multi-variate Gaussians for L-functions and applications to zeros
RMAW04 28th June 2004
09:30 to 10:25
Quantum vesus classical fluctuations on the modular surface

In joint work with W.Luo we have computed the variance of the fluctuations of a quantum observable on the modular surface. It corresponds to the classical fluctuations of the observable after insertion of a subtle arithmetic correction factor.

RMAW04 28th June 2004
11:00 to 11:40
S De Bievre Long time propagation of coherent states under perturbed cat map dynamics

I will describe recent work with J.M. Bouclet (Lille) on the propagation of coherent states up to times logarithmic in hbar under quantized perturbed cat maps. We show that, for long enough times, the quantum evolution equidistributes the coherent states throughout phase space. The proof requires a good control on the error term in the Egorov theorem on the one hand and on the classical rate of mixing on the other. This generalizes to perturbed cat maps a result obtained previously with F. Bonechi.

RMAW04 28th June 2004
11:50 to 12:30
Random matrix theory and entanglement in quantum spin chains

We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians --- those that are related to quadratic forms of Fermi operators --- between the first $N$ spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as $N\rightarrow\infty$. This is shown to grow logarithmically with $N$. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlev\'e type. In some cases these solutions can be evaluated to all orders using recurrence relations.

RMAW04 28th June 2004
14:30 to 15:25
Evolution and constrains on scarring for (perturbed) cat maps

We consider quantized cat maps on the 2-dimensional torus, as well as their nonlinear perturbations. We first analyze the evolution up to the Ehrenfest time of states localized around a periodic point, showing a transition to equidistribution. Using this transition, we obtain constraints on the localization properties of eigenstates around periodic orbits. The analysis is much simpler in the unperturbed case, where one uses the algebraic properties of the map. Besides, the constraints we obtain are known to be sharp only for the unperturbed case.

RMAW04 28th June 2004
16:00 to 16:40
C Hughes On the number of lattice points in a thin annulus

We count the number of integer lattice points in an annulus of inner-radius $t$ and outer-radius $t+\rho$. If $\rho \to 0$ sufficiently slowly then the distribution of this counting function as $t\to\infty$ weakly converges to the normal distribution.

RMAW04 28th June 2004
16:50 to 17:30
N Anantharaman The ``Quantum unique ergodicity" problem for anosov geodesic flows: an approach by entropy

Let $M$ be compact, negatively curved Riemannian manifold, and let $(\psi_n)$ be an orthonormal basis of eigenfunctions of the Laplacian on $M$. The Quantum Unique Ergodicity problem concerns the behaviour of the sequence of probability measures $|\psi_n(x)|^2 dx$ on $M$, or, more precisely, of their "microlocal" lifts to the tangent bundle $TM$. The limits of convergent subsequences must be invariant probability measures of the geodesic flow (sometimes called "quantum invariant measures"), and it is known that a very large subsequence converges to the Liouville measure on the unit tangent bundle. A conjecture of Rudnick and Sarnak says that this should actually be the only possible limit. E. Lindenstrauss proved the conjecture recently, in the case when $M$ is an arithmetic surface (of constant negative curvature) and $(\psi_n)$ is a common basis of eigenfunctions for the Laplacian and the Hecke operators. However, very little is known in the non-arithmetic case.

In the general case of an Anosov geodesic flow, I present an attempt to bound from below the metric entropy of "quantum invariant measures". I actually prove the following: if the $L^p$ norms of the $\psi_n$s do not grow too fast with $n$, then the corresponding quantum invariant measures cannot be entirely carried on a set of zero topological entropy.

RMAW04 29th June 2004
09:30 to 10:25
Semiclassical evidence for universal spectral correlations in quantum chaos

Almost all quantum systems which are chaotic in their classical limit exhibit universality when statistical distributions of energy levels are evaluated. Spectral correlations are found to agree with those between eigenvalues of random matrices. We report on recent progress in semiclassical methods that provide a theoretical basis for the connection between quantum chaos and random matrix theory.

RMAW04 29th June 2004
11:00 to 11:40
R Schubert Propagation of wavepackets for large times

We study the semiclassical propagation of a class of wavepackets for large times on manifolds of negative curvature. The time evolution is generated by the Laplace-Beltrami operator and the wavepackets considered are Lagrangian states. The principal result is that these wavepackets become weakly equidistributed in the joint limit $\hbar\to 0$ and $t\to\infty$ with $t<<|\ln \hbar|$. The main ingredient in the proof is hyperbolicity and mixing of the geodesic flow.

RMAW04 29th June 2004
11:50 to 12:30
On the distribution of matrix elements for the quantum cat map

For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable. We will study the fluctuations of the matrix elements for the desymmetrized quantum cat map and present a conjecture for the distribution of the normalized matrix elements, namely that their distribution is that of a certain weighted sum of traces of independent matrices in SU(2). This is in contrast to generic chaotic systems where the distribution is expected to be Gaussian. We will show that the second and fourth moment of the distribution agree with the conjecture, and also present some numerical evidence.

Related Links

RMAW04 29th June 2004
14:30 to 15:25
Quantum chaos on locally symmetric spaces

I will discuss a proof of QUE, w.r.t. the basis of Hecke eigenforms, for certain locally symmetric spaces of rank > 1.

RMAW04 29th June 2004
16:00 to 16:40
On the remiainder in weyl's law for heisenberg manifolds

We examine the error term in Weyl's law for the distribution of Laplace eigenvalues for Heisenberg manifolds equipped with a left-invariant metric. We formulate conjectures on the optimal size and evidence for the conjectures. We explain the analogies with the classical Dirichlet divisor problem in analytic number theory. This program has been developped jointly with D. Chung, M. Khosravi and J. Toth.

Related Links

RMAW04 29th June 2004
16:50 to 17:30
S Müller Semiclassical foundation of universality in quantum chaos

We sketch the semiclassical core of a proof of the so-called Bohigas-Giannoni-Schmit conjecture: A dynamical system with full classical chaos has a quantum energy spectrum with universal fluctuations on the scale of the mean level spacing. We show how in the semiclassical limit all system specific properties fade away, leaving only ergodicity, hyperbolicity, and combinatorics as agents determining the contributions of pairs of classical periodic orbits to the quantum spectral form factor. The small-time form factor is thus reproduced semiclassically. Bridges between classical orbits and (the non-linear sigma model of) quantum field theory are built by revealing the contributing orbit pairs as topologically equivalent to Feynman diagrams.

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RMAW04 30th June 2004
09:30 to 10:25
Classical dynamics of billiards in rational polygons

This talk is a survey of some recent developments in the field.

RMAW04 30th June 2004
11:00 to 11:40
The triangle map: a model for quantum chaos

We intend to discuss some recent results concerning classical and semiclassical properites of a particular weakly chaotic discrete dynamical system.

RMAW04 30th June 2004
11:50 to 12:30
Eigenfunction statistics for star graphs

I will review some recent results, obtained in collaborations with Gregory Berkolaiko, Jens Marklof and Brian Winn, relating to the title of the talk.

RMAW04 1st July 2004
09:30 to 10:25
A central limit theorem for the spectrum of the modular domain

We study the fluctuations in the discrete spectrum of the hyperbolic Laplacian for the modular domain using smooth counting functions. We show that in a certain regime, these have Gaussian fluctuations.

RMAW04 1st July 2004
11:00 to 11:40
Subconvexity of L-functions and the uniqueness principle

We consider the triple L-function $L(1/2,f\times g\times \phi_i)$ for fixed Maass forms f and g as the eigenvalue of $\phi_i$ goes to infinity.

We deduce a subconvexity bound for this L-function from the uniqueness principle in representation theory and from simple geometric properties of the corresponding invariant functional. Joint with J. Bernstein.

RMAW04 1st July 2004
11:50 to 12:30
On distribution of zeros of Heine-Stieltjes polynomials

We introduce Heine-Stieltjes polynomials, describe classical results of Heine, Stieltjes, Van Vleck and Shah on the distribution of their zeros as well as more recent asymptotic results, both in the semiclassical and thermodynamic regime.

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RMAW04 1st July 2004
14:30 to 15:25
Complex zeros of real ergodic eigenfunctions

A well-known problem in geometry of eigenfunctions of Laplacians on Riemannian manifolds is to determine how the nodal hypersurface (zero set) is asymptotically distributed as the eigenvalue tends to infinity. The random wave model predicts that the normalized measure of integration over the nodal hypersurface tends to the volume measure on the manifold. My talk is a preliminary report on the distribution of complex zeros of analytic continuations of eigenfunctions of real analytic Riemannian manifolds with ergodic geodesic flow. We describe how the complex nodal hypersurfaces are distributed in the cotangent bundle. The (perhaps surprising) result is that the complex zeros concentrate around the real ones.

RMAW04 1st July 2004
16:00 to 16:40
Energy asymptotics for gaudin spin chains

I will discuss recent work (joint with M. Min-Oo) on partition function asymptotics for the integrable Gaudin spin chains in various thermodynamic regimes.

RMAW04 2nd July 2004
09:30 to 10:25
F Steiner The cosmic microwave background and the shape of the Universe

The anisotropy of the cosmic microwave background (CMB) is analysed in nearly flat hyperbolic universes possessing a non-trivial topology with a fundamental cell which is stretched out into an infinitely long horn. It is shown that the horned topology does not lead to a flat spot in the CMB sky maps in the direction of the horn as predicted by Levin, Barrow, Bunn, and Silk.Two particular topologies are discussed in detail: the Sokolov-Starobinsky model having an infinite spatial volume, and the Picard orbifold which has a finite volume. It is demonstrated that the recent observations of the "Wilkinson Microwave Anisotropy Probe" (WMAP) hint that our Universe may be shaped like the Picard space.

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RMAW04 2nd July 2004
11:00 to 11:40
Zeros of the derivative of a selberg zeta function

In this talk, we will study the distribution of non-trivial zeros of Selberg zeta functions on cofinite hyperbolic surfaces, in particular obtain the asymptotic formula for the zero density with bounded height, which is similar to the well-known Weyl law. Then we will relate the distribution of the zeros to the issue of bounding the multiplicity of Laplacian eigenvalues.

RMAW04 2nd July 2004
11:50 to 12:30
A Strombergsson Numerical computations with the trace formula and the selberg eigenvalue conjecture

I will report on some numerical computations with the trace formula on congruence subgroups of the modular group. In particular I will discuss how to check numerically the validity of the Selberg eigenvalue conjecture for specific congruence subgroups.

This is joint work with Andrew Booker.

RMAW04 2nd July 2004
14:30 to 15:25
Granular bosonization (or Fyodorov meets SUSY)

Random matrix methods can be roughly divided into two categories: methods which rely on having the joint probability density of the eigenvalues in closed form, and others which don't. Supersymmetry methods belong to the latter category.

The supersymmetry method used in the theory of disordered metals goes back to Schaefer and Wegner (1980). Quite recently, Fyodorov has proposed a related but different method, which computes averages of inverse characteristic polynomials for granular systems or random matrices with a hierarchical structure.

In this talk Fyodorov's method is reviewed, and it is shown how to generalize it to the case of characteristic polynomials (where it amounts to a form of bosonization) and to the case of ratios of such polynomials (the supersymmetric variant). Some applications to granular systems are presented.

RMAW04 2nd July 2004
16:00 to 16:40
The double Riemann zeta function

A double zeta function is defined as a function having zeros at sum of zeros of an original zeta function. The aim of this talk is to construct the double Riemann zeta function, and express it as a double Euler product which is a product over pairs of prime numbers.

RMAW04 2nd July 2004
16:50 to 17:30
A Gamburd Expander graphs, random matrices and quantum chaos

A basic problem in the theory of expander graphs, formulated by Lubotzky and Weiss, is to what extent being an expander family for a family of Cayley graphs is a property of the groups alone, independent of the choice of generators. While recently Alon, Lubotzky and Wigderson constructed an example demonstrating that expansion is not in general a group property, the problem is open for "natural" families of groups. In particular for SL(2, p) numerical experiments indicate that it might be an expander family for "generic" choices of generators (Independence Conjecture).

A basic conjecture in Quantum Chaos, formulated by Bohigas, Giannoni, and Shmit, asserts that the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of a typical member of the appropriate ensemble of random matrices. Both conjectures can be viewed as asserting that a deterministically constructed spectrum "generically" behaves like the spectrum of a large random matrix: "in the bulk" (Quantum Chaos Conjecture) and at the "edge of the spectrum" (Independence Conjecture). After explaining this approach in the context of the spectra of random walks on groups, we review some recent related results and numerical experiments.

RMA 8th July 2004
14:30 to 15:30
Uncertainty principles in arithmetic
RMAW01 8th July 2004
16:00 to 17:00
Arithmetic progressions of primes
RMAW05 12th July 2004
11:00 to 11:45
C Hughes Mollified \& amplified moments: Some new theorems \& conjectures

Two of the most important areas in analytic number theory concern counting the number of zeros of zeta functions on and off the line, and in beating subconvexity bounds. Both types of results can be obtained from knowing moments of the zeta function multiplied by a Dirichlet polynomial. In this talk we present an asymptotic formula for the fourth moment of the zeta function multiplied by a Dirichlet polynomial, and conjecture a formula for general moments.

RMAW05 12th July 2004
12:00 to 12:30
Primes \& pair correlation of zeros

After a review of older work on this topic, some new results obtained jointly with Soundararajan will be described. These concern higher moments of the error term for the number of primes in a short interval.

RMAW05 12th July 2004
14:30 to 15:00
On the moments of Hecke series at central points
RMAW05 12th July 2004
15:00 to 15:30
The twelth moment of central values of Hecke series
RMAW05 12th July 2004
16:00 to 16:45
Testing random matrix theory vs the zeta zeros

I will give a tutorial on methods of testing predictions of random matrix theory on data. There is some nice math (symmetric function theory) and some subtlety (the level repulsions lead to correlated data and need cutting edge tools such as the block bootstrap). This is joint work with Marc Coram.

RMAW05 12th July 2004
17:00 to 17:30
Automorphic summation formulae and moments of zeta

The strong parallel between conjectural asymptotics of the 2n-th moment of zeta (Conrey, Farmer, Keating, Rubinstein and Snaith) with a ``constant term'' of an Eisenstein series on GL(2n) will be reviewed. For the second moment, the parallel is explained by the Voronoi-Oppenheim summation formula. For larger n, divisor functions of lattices will be defined and a pleasant new Voronoi-type summation formula will be proved for the lattice divisor functions, making use of Bessel functions associated with the Shalika-Kirillov model of a degenerate principal series representation of GL(2n,R).

RMAW05 12th July 2004
17:30 to 18:00
I Smolyarenko Parametric RMT, discrete symmetries, \& cross-correlations between zeros of L-functions

I will describe numerical and analytical results on cross-correlations between zeros of different L-functions. By analogy with parametric spectral correlations in random matrix theory and in dynamical systems, these cross-correlations can be used to establish the concept of a "distance" in the space of (conjectural) generalised Riemann operators, and to gain some insight into their overall structure.

RMAW05 13th July 2004
09:30 to 10:20
A survey of elliptic curves

This talk will be a fairly down-to-earth survey of the theory of elliptic curves, with special emphasis on stating the Birch and Swinnerton-Dyer conjecture and explaining the various invariants that enter into it, and in particular how this turns out to be related to Random Matrix Theory.

RMAW05 13th July 2004
11:00 to 11:45
Extreme values \& moments of L-functions
RMAW05 13th July 2004
12:00 to 12:30
A Booker Poles of L-functions \& the converse theorem
RMAW05 13th July 2004
14:30 to 15:20
Perspectives on L functions and spectral theory
RMAW05 13th July 2004
16:00 to 16:45
Negative moments

I will describe recent results, obtained with Peter Forrester, concerning the negative moments of the characteristic polynomials of random matrices and some implications in number theory.

RMAW05 13th July 2004
20:30 to 18:00
Escher and the Droste effect

M.C. Escher, the graphic artist famous for mathematical patterns and optical illusions, left a blank spot in the middle of his 1956 lithograph, "Print Gallery". Escher signed his name there, instead of completing the center of this picture of curved and whirling buildings. In 2002, a team of mathematicians, computer programmers, and artists used techniques from advanced mathematics to figure out how Escher might have completed the picture. The team was led by Hendrik Lenstra, who will reveal the secrets behind the mysterious blank space during this talk. The presentation and explanation of the mystery will include a number of beautiful images and animations.

Hendrik Lenstra is Professor of Mathematics at the Universiteit Leiden, the Netherlands. Dr Lenstra is a world-renowned mathematician who is known for the clarity and wit of his lectures. His presentation will be aimed at a general audience and should appeal to anyone interested in art, mathematics, or the intersection of the two subjects.

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RMAW05 14th July 2004
09:30 to 10:20
Introduction to function fields

I will give an overview, intended for non-experts, of some arithmetic aspects of function fields of curves over finite fields.

RMAW05 14th July 2004
11:00 to 11:45
Random matrix theory \& life over finite fields

We will give an introductory survey of some of the relations between random matrix theory and various diophantine questions over finite fields.

RMAW05 14th July 2004
12:00 to 12:30
Symmetry beyond root numbers: a GL(6) example (joint with S Miller)

Going against the "folklore" belief that even orthogonal families arise splitting a full orthogonal family by sign, we show that the lone-standing family {L(s,g x Sym2(f)} (where g is a fixed Hecke-Maass form and f varies over holomorphic modular forms of level 1) has SO(even) symmetry. Thus, the theory of symmetry types is not merely about root numbers (sign of the functional equation). The family above is connected with the relation between classical and quantum fluctuations of observables in the modular surface by work of Luo and Sarnak.

RMAW05 15th July 2004
09:30 to 10:20
F Rodriguez-Villegas Computing twisted central values of L-functions

I will describe the various techniques available for computing central values of quadratic twists of a given L-function associated to a modular form.

RMAW05 15th July 2004
11:00 to 11:45
The geometry of zeros

For both characteristic polynomials and L-functions, we will consider the relationship between zeros, large values, and zeros of the derivative. We will discuss the maximal order of the zeta-function, and we will describe some unsolved problems in random matrix theory that could illuminate difficult questions in number theory.

RMAW05 15th July 2004
12:00 to 12:30
Zeros of random polynomials \& linear combinations of random characteristic polynomials

In this talk, we shall give results about the asymptotic behavior of roots of random polynomials in the plane. We then specialize to the case of sums of characteristic polynomials of random unitary matrices: we prove an analogue of a result by Bombieri and Hejhal about the zeros on the critical line for linear combination of L-functions.

RMAW05 15th July 2004
14:30 to 15:20
Experiments in number theory \& random matrix theory
RMAW05 15th July 2004
16:00 to 16:45
A new statistical model of the Riemann zeta function

The characteristic polynomial models of the Riemann zeta function and other L-functions have allowed us to predict answers to a variety of questions previously considered intractable. However, these powerful models have contained no arithmetical information, which generally has to be introduced in an ad hoc manner. I will present a new model for the zeta function developed with C. Hughes and J. Keating that overcomes this difficulty. I will illustrate its use by calculating moments of the Riemann zeta function and estimating the maximal order of the zeta function on the critical line.

RMAW05 16th July 2004
09:30 to 10:20
A Gamburd Applications of symmetric functions theory to random matrices
RMAW05 16th July 2004
11:00 to 11:45
Ratios of random characteristic polynomials from supersymmetry
RMAW05 16th July 2004
12:00 to 12:30
Vanishing of L-functions of elliptic curves over number fields
RMAW05 16th July 2004
14:30 to 15:20
Multiple Dirichlet series, an historical survey

Multiple Dirichlet series (L-functions of several complex variables) are Dirichlet series in one complex variable whose coefficients are again Dirichlet series in other complex variables. These series arise naturally in the theory of moments of zeta and L-functions. It was found recently by Diaconu-Goldfeld-Hoffstein that the moment conjectures of random matrix theory, such as the Keating-Snaith conjecture, would follow if certain multiple Dirichlet series had meromorphic continuation to a a particular tube domain.

We shall present an introduction to some of the basic definitions and techniques of this theory as well as a survey of some of the results that have been obtained by this method. These include applications to moments of L-functions, Fermat's last theorem, classification theory via Dynkin diagrams, and analysis of natural constructions as inner products of automorphic forms on GL(n).

RMAW05 16th July 2004
16:00 to 16:45
Ratios of zeta functions \& characteristic polynomials

This talk will describe the heuristic method of calculating averages of ratios of zeta and L-functions used by Conrey, Farmer and Zirnbauer to generalize conjectures of Farmer. It will include applications of these ratio formulae and a new method devised with Conrey and Forrester will be discussed for calculating the analogous random matrix quantities.

RMAW05 16th July 2004
17:00 to 17:30
A Perelli The Selberg class of L-functions: non-linear twists

In a paper published in 1992, Selberg introduced an axiomatic class of L-functions (now called the Selberg class S) and raised several very interesting problems. In particular, Selberg raised the problem of classifying the L-functions in S. In this talk we first review the results on the classification problem. Such results depend on the analytic properties of the linear twists of the L-functions. Then we introduce certain non-linear twists and present some recent work (joint with J.Kaczorowski) on their analytic properties and applications.

RMAW05 16th July 2004
17:30 to 17:50
G Molteni Bounds at s=1 for an axiomatic class of L-functions

When the Ramanujan hypothesis about the Dirichlet coefficients of a generic L-function is assumed, it is quite easy to prove upper-bounds of type L(1)<< R^c, for every c>0, where R is a parameter related to the functional equation of L. We show how to prove the same bound when the Ramanujan hypothesis is replaced by a much weaker assumption and L has Euler product of polynomial type. As a consequence, we obtain an upper bound of this type for every cuspidal automorphic GL(n) L-function, unconditionally. We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the symmetric cube of a Maass form.

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