Videos and presentation materials from other INI events are also available.
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 padic 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/SwinnertonDyer 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 TateShafarevich groups  
RMAW01 
9th February 2004 15:30 to 16:15 
M Rubinstein  Moments, Lvalues & Ranks  
RMAW01 
9th February 2004 16:45 to 17:30 
Vanishing of Lfunctions 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 Lfunctions  
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 zetafunction I Lecture 1: Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model Lecture 2: Open questions on primes Recent achievements of prime number theory The Riemann Zetafunction Euler product Analytic continuation and functional equation (via theta function) Lecture 3: Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T) Lecture 4: N(T) and S(T) (continued) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

RMAW02 
29th March 2004 11:30 to 12:30 
Y Fyodorov  Gaussian ensembles of random matrices I  
RMAW02 
29th March 2004 14:00 to 15:00 
Prime number theory \& the Riemann zetafunction II Lecture 1: Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model Lecture 2: Open questions on primes Recent achievements of prime number theory The Riemann Zetafunction Euler product Analytic continuation and functional equation (via theta function) Lecture 3: Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T) Lecture 4: N(T) and S(T) (continued) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

RMAW02 
29th March 2004 15:30 to 16:30 
Y Fyodorov  Gaussian ensembles of random matrices II  
RMAW02 
29th March 2004 16:30 to 17:30 
Artin Lfunctions  
RMAW02 
30th March 2004 09:00 to 10:00 
Prime number theory & the Riemann zetafunction III Lecture 1: Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model Lecture 2: Open questions on primes Recent achievements of prime number theory The Riemann Zetafunction Euler product Analytic continuation and functional equation (via theta function) Lecture 3: Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T) Lecture 4: N(T) and S(T) (continued) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

RMAW02 
30th March 2004 10:00 to 11:00 
Elliptic curves  
RMAW02 
30th March 2004 11:30 to 12:30 
Y Fyodorov  Gaussian ensembles of random matrices III  
RMAW02 
30th March 2004 14:30 to 15:30 
Prime number theory & the Riemann zetafunction IV Lecture 1: Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model Lecture 2: Open questions on primes Recent achievements of prime number theory The Riemann Zetafunction Euler product Analytic continuation and functional equation (via theta function) Lecture 3: Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T) Lecture 4: N(T) and S(T) (continued) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

RMAW02 
30th March 2004 16:00 to 17:00 
Y Fyodorov  Gaussian ensembles of random matrices IV  
RMAW02 
30th March 2004 17:00 to 18:00 
Pair correlation of zeros of the Riemann zetafunction and prime numbers I The first lecture will cover Montgomery's 1973 paper where the pair correlation and GUE conjecture were first made, together with some applications. The second and third lecture will cover the connections between zeros and prime numbers, and include further methods for examining the distribution of primes. The fourth lecture will give a brief introduction to Selberg's theory of the distribution of values of log (zeta) and S(t). Related Links


RMAW02 
31st March 2004 09:00 to 10:00 
Pair correlation of zeros of the Riemann zetafunction and prime numbers II The first lecture will cover Montgomery's 1973 paper where the pair correlation and GUE conjecture were first made, together with some applications. The second and third lecture will cover the connections between zeros and prime numbers, and include further methods for examining the distribution of primes. The fourth lecture will give a brief introduction to Selberg's theory of the distribution of values of log (zeta) and S(t). Related Links


RMAW02 
31st March 2004 10:00 to 11:00 
Heuristic derivation of the npoint correlation function for the Riemann zeros I  
RMAW02 
31st March 2004 11:30 to 12:30 
Modular forms  
RMAW02 
1st April 2004 09:00 to 10:00 
Prime number theory \& the Riemann zetafunction V Lecture 1: Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model Lecture 2: Open questions on primes Recent achievements of prime number theory The Riemann Zetafunction Euler product Analytic continuation and functional equation (via theta function) Lecture 3: Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T) Lecture 4: N(T) and S(T) (continued) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

RMAW02 
1st April 2004 10:00 to 11:00 
Pair correlation of zeros of the Riemann zetafunction and prime numbers III The first lecture will cover Montgomery's 1973 paper where the pair correlation and GUE conjecture were first made, together with some applications. The second and third lecture will cover the connections between zeros and prime numbers, and include further methods for examining the distribution of primes. The fourth lecture will give a brief introduction to Selberg's theory of the distribution of values of log (zeta) and S(t). Related Links


RMAW02 
1st April 2004 11:30 to 12:30 
Y Fyodorov  Gaussian ensembles of random matrices V  
RMAW02 
1st April 2004 14:00 to 15:00 
Heuristic derivation of the npoint correlation function for the Riemann zeros II  
RMAW02 
1st April 2004 15:30 to 16:30 
Pair correlation of zeros of the Riemann zetafunction and prime numbers IV The first lecture will cover Montgomery's 1973 paper where the pair correlation and GUE conjecture were first made, together with some applications. The second and third lecture will cover the connections between zeros and prime numbers, and include further methods for examining the distribution of primes. The fourth lecture will give a brief introduction to Selberg's theory of the distribution of values of log (zeta) and S(t). Related Links


RMAW02 
1st April 2004 16:30 to 17:30 
Lfunctions over functions fields  
RMAW02 
2nd April 2004 09:00 to 10:00 
Y Fyodorov  Gaussian ensembles of random matrices VI  
RMAW02 
2nd April 2004 10:00 to 11:00 
Prime number theory \& the Riemann zetafunction VI Lecture 1: Unique Factorization Theorem Infinitude of primes Statement of PNT Cramer model Failure of Cramer model Lecture 2: Open questions on primes Recent achievements of prime number theory The Riemann Zetafunction Euler product Analytic continuation and functional equation (via theta function) Lecture 3: Analytic continuation and functional equation (continued) Hadamard product and its logarithmic derivative N(T) and S(T) Lecture 4: N(T) and S(T) (continued) Nonvanishing on the 1line Proof of PNT Lecture 5: Proof of PNT (continued) Weil type Explicit formulae Lecture 6: Characters Dirichlet Lfunctions Prerequisites: Undergrad complex analysis Prpoerties of the gamma function Undergrad algebra (Z is a UFD) Recommended text: Davenport, Multiplicative Number Theory 

RMAW02 
2nd April 2004 11:30 to 12:30 
Heuristic derivation of the npoint correlation function for the Riemann zeros III  
RMAW02 
2nd April 2004 14:00 to 15:00 
OG Bohigas 
Compund nucleus resonances, random matrices, quantum chaos Wigner introduced random matrices in physics when searching for a guiding principle to understand properties of the compound nucleus resonances. At the end the experimental observations turned out to be remarkably consistent with random matrix theory predictions. Could random matrix theory be justified in dynamical terms? To answer this question deep connections between quantum bahaviour of classically chaotic systems (quantum chaos) and random matrices have been established. Open problems still remain. Some highlights of this long excursion, covering more than fourty years, will be given. 

RMAW02 
2nd April 2004 15:30 to 16:30 
Quantum chaology and zeta As requested by the organisers the talk will consist of my reminiscences of how the different intellectual streams of quantum chaology and Riemannology became intermingled, with benefit to both. 

RMAW02 
3rd April 2004 10:00 to 11:00 
Mean values & zeros of the zeta function  
RMAW02 
3rd April 2004 11:30 to 12:30 
Toeplitz determinants & connections to random matrices I Lecture1: This lecture will contain an introduction of Toeplitz Operators, Hankel Operators, and finite Toeplitz matrices. We will give a simple proof of the Strong Szego Limit Theorem which describes an asymptotic formula for determinants of Toeplitz matrices with smooth symbols. Lecture2: This lecture will show how to extend the Szego Limit Theorem to symbols with discontinuities, the class of FisherHartwig symbols. Lecture3: The final lecture will show the connection of the limit theorems to results about the distribution functions for random matrix ensembles. If time permits, Laguerre ensembles will be considered and also the analogues for WienerHopf opeators. 

RMAW02 
3rd April 2004 13:30 to 14:30 
Spacing distributions for random matrix ensembles I The calculation of eigenvalue spacing distributions for classical random matrix ensembles with unitary symmetry is intimately related to the theory of integrable systems and Painleve' equations. These theories provide the characterization of spacing distributions as solutions of nonlinear equations solvable in terms of Painleve' transcendents. Two approaches to achieve this goal will be detailed. One approach makes use of function theoretic properties of Fredholm determinants. The other proceeds via the theory of Painleve' systems. Related Links


RMAW02 
5th April 2004 09:00 to 10:00 
RMT moment calculations I Lecture 1: Introduction to the CUE and averages over other classical compact groups Lecture 2: moments and value distribution of CUE characteristic polynomials; comparison with the moments and value distribution of the Riemann zeta function Lecture 3: vanishing of Lfunction central values and RMT predictions 

RMAW02 
5th April 2004 10:00 to 11:00 
B Conrey  Statistics of lowlying zeros of Lfunction and random matrix theory I  
RMAW02 
5th April 2004 11:30 to 12:30 
Low moments of the Riemann zeta function  
RMAW02 
5th April 2004 14:00 to 15:00 
RMT moment calculations II Lecture 1: Introduction to the CUE and averages over other classical compact groups Lecture 2: moments and value distribution of CUE characteristic polynomials; comparison with the moments and value distribution of the Riemann zeta function Lecture 3: vanishing of Lfunction central values and RMT predictions 

RMAW02 
5th April 2004 15:30 to 16:30 
C Hughes 
Derivatives of the Riemann zeta function In this lecture we will look at moments of the derivative of the Riemann zeta function, and how random matrix theory can help predict their asymptotic behaviour. Applications to estimating the size of large gaps between the zeros will also be discussed. 

RMAW02 
5th April 2004 16:30 to 17:30 
M Rubinstein  Computational methods for Lfunctions I  
RMAW02 
6th April 2004 09:00 to 10:00 
E Basor 
Toeplitz determinants \& connections to random matrices II Lecture1: This lecture will contain an introduction of Toeplitz Operators, Hankel Operators, and finite Toeplitz matrices. We will give a simple proof of the Strong Szego Limit Theorem which describes an asymptotic formula for determinants of Toeplitz matrices with smooth symbols. Lecture2: This lecture will show how to extend the Szego Limit Theorem to symbols with discontinuities, the class of FisherHartwig symbols. Lecture3: The final lecture will show the connection of the limit theorems to results about the distribution functions for random matrix ensembles. If time permits, Laguerre ensembles will be considered and also the analogues for WienerHopf opeators. 

RMAW02 
6th April 2004 10:00 to 11:00 
Spacing distributions for random matrix ensembles II The calculation of eigenvalue spacing distributions for classical random matrix ensembles with unitary symmetry is intimately related to the theory of integrable systems and Painleve' equations. These theories provide the characterization of spacing distributions as solutions of nonlinear equations solvable in terms of Painleve' transcendents. Two approaches to achieve this goal will be detailed. One approach makes use of function theoretic properties of Fredholm determinants. The other proceeds via the theory of Painleve' systems. Related Links


RMAW02 
6th April 2004 11:30 to 12:30 
B Conrey  Statistics of lowlying zeros of Lfunction and random matrix theory II  
RMAW02 
6th April 2004 14:00 to 15:00 
M Rubinstein  Computational methods for Lfunctions II  
RMAW02 
6th April 2004 15:30 to 16:30 
B Conrey  Statistics of lowlying zeros of Lfunction and random matrix theory III  
RMAW02 
6th April 2004 16:30 to 17:30 
Toeplitz determinants \& connections to random matrices III Lecture1: This lecture will contain an introduction of Toeplitz Operators, Hankel Operators, and finite Toeplitz matrices. We will give a simple proof of the Strong Szego Limit Theorem which describes an asymptotic formula for determinants of Toeplitz matrices with smooth symbols. Lecture2: This lecture will show how to extend the Szego Limit Theorem to symbols with discontinuities, the class of FisherHartwig symbols. Lecture3: The final lecture will show the connection of the limit theorems to results about the distribution functions for random matrix ensembles. If time permits, Laguerre ensembles will be considered and also the analogues for WienerHopf opeators. 

RMAW02 
7th April 2004 09:00 to 10:00 
Mean values of Dirichlet polynomials \& applications  
RMAW02 
7th April 2004 10:00 to 11:00 
B Conrey  Statistics of lowlying zeros of Lfunction and random matrix theory IV  
RMAW02 
7th April 2004 11:30 to 12:30 
M Rubinstein  Computational methods for Lfunctions III  
RMAW02 
7th April 2004 14:00 to 15:00 
C Hughes 
A new model for the Riemann zeta function Random matrix theory (RMT) has been very successul at modeling the zeros of the zeta function. A recent conjecture of Keating and Snaith uses RMT to conjecture the asymptotic form of moments of the Riemann zeta function, but the conjecture requires an adhoc addition from primes to fit known results. In this lecture a new model for the zeta function will be presented, where it is writen as a partial Euler product times a partial Hadamard product. This model enables us to rederive the KeatingSnaith conjecture with both the prime contribution and the random matrix contribution appearing naturally. The research presented in this lecture is joint with Jon Keating and Steve Gonek. 

RMAW02 
7th April 2004 15:30 to 16:30 
Spacing distributions for random matrix ensembles III The calculation of eigenvalue spacing distributions for classical random matrix ensembles with unitary symmetry is intimately related to the theory of integrable systems and Painleve' equations. These theories provide the characterization of spacing distributions as solutions of nonlinear equations solvable in terms of Painleve' transcendents. Two approaches to achieve this goal will be detailed. One approach makes use of function theoretic properties of Fredholm determinants. The other proceeds via the theory of Painleve' systems. Related Links


RMAW02 
8th April 2004 09:00 to 10:00 
C Hughes 
MockGaussian behaviour MockGaussian behaviour is when a smooth counting function (or linear statistic) has its first few moments equal to the moments of a Gaussian distribution, even though it is not a normal distribution. In this lecture we will see that this behaviour holds eigenvalues of random matrices, and analogously for the zeros of the Riemann zeta function and other Lfunctions. The research presented in this lecture is joint with Zeev Rudnick. Related Links


RMAW02 
8th April 2004 10:00 to 11:00 
Families \& conjectures for moments of Lfunctions  
RMAW02 
8th April 2004 11:30 to 12:30 
RMT moment calculations III Lecture 1: Introduction to the CUE and averages over other classical compact groups Lecture 2: moments and value distribution of CUE characteristic polynomials; comparison with the moments and value distribution of the Riemann zeta function Lecture 3: vanishing of Lfunction central values and RMT predictions 

RMA 
15th April 2004 14:30 to 15:30 
The partition function p(n) and the manybody density of states  
RMA 
15th April 2004 16:00 to 17:00 
Applications and generalisations of FisherHartwig 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 multiparticle lattice systems  
RMA 
27th May 2004 16:00 to 17:00 
Lowlying zeroes of quadratic Lfunctions  
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 Lfunctions  
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  Multivariate Gaussians for Lfunctions 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 spinchain 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 FisherHartwig 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 conformalfieldtheoretic 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 2dimensional 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 innerradius $t$ and outerradius $t+\rho$. 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 nonarithmetic 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 LaplaceBeltrami 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 leftinvariant 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 socalled BohigasGiannoniSchmit 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 smalltime form factor is thus reproduced semiclassically. Bridges between classical orbits and (the nonlinear sigma model of) quantum field theory are built by revealing the contributing orbit pairs as topologically equivalent to Feynman diagrams. Related Links


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 Lfunctions and the uniqueness principle We consider the triple Lfunction $L(1/2,f\times g\times \phi_i)$ for fixed Maass forms f and g as the eigenvalue of $\phi_i$ goes to infinity. We deduce a subconvexity bound for this Lfunction 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 HeineStieltjes polynomials We introduce HeineStieltjes 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. Related Links


RMAW04 
1st July 2004 14:30 to 15:25 
Complex zeros of real ergodic eigenfunctions A wellknown 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. MinOo) 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 nontrivial 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 SokolovStarobinsky 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. Related Links 

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 nontrivial 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 wellknown 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 2nth 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 VoronoiOppenheim summation formula. For larger n, divisor functions of lattices will be defined and a pleasant new Voronoitype summation formula will be proved for the lattice divisor functions, making use of Bessel functions associated with the ShalikaKirillov 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, \& crosscorrelations between zeros of Lfunctions I will describe numerical and analytical results on crosscorrelations between zeros of different Lfunctions. By analogy with parametric spectral correlations in random matrix theory and in dynamical systems, these crosscorrelations 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 downtoearth survey of the theory of elliptic curves, with special emphasis on stating the Birch and SwinnertonDyer 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 Lfunctions  
RMAW05 
13th July 2004 12:00 to 12:30 
A Booker  Poles of Lfunctions \& 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 worldrenowned 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. Related Links


RMAW05 
14th July 2004 09:30 to 10:20 
Introduction to function fields I will give an overview, intended for nonexperts, 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 lonestanding family {L(s,g x Sym2(f)} (where g is a fixed HeckeMaass 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 RodriguezVillegas 
Computing twisted central values of Lfunctions I will describe the various techniques available for computing central values of quadratic twists of a given Lfunction associated to a modular form. 

RMAW05 
15th July 2004 11:00 to 11:45 
The geometry of zeros For both characteristic polynomials and Lfunctions, we will consider the relationship between zeros, large values, and zeros of the derivative. We will discuss the maximal order of the zetafunction, 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 Lfunctions. 

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 Lfunctions 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 Lfunctions of elliptic curves over number fields  
RMAW05 
16th July 2004 14:30 to 15:20 
Multiple Dirichlet series, an historical survey Multiple Dirichlet series (Lfunctions 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 Lfunctions. It was found recently by DiaconuGoldfeldHoffstein that the moment conjectures of random matrix theory, such as the KeatingSnaith 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 Lfunctions, 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 Lfunctions 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 Lfunctions: nonlinear twists In a paper published in 1992, Selberg introduced an axiomatic class of Lfunctions (now called the Selberg class S) and raised several very interesting problems. In particular, Selberg raised the problem of classifying the Lfunctions 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 Lfunctions. Then we introduce certain nonlinear 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 Lfunctions When the Ramanujan hypothesis about the Dirichlet coefficients of a generic Lfunction is assumed, it is quite easy to prove upperbounds 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) Lfunction, unconditionally. We employ these results to obtain Siegeltype lower bounds for twists by Dirichlet characters of the symmetric cube of a Maass form. 