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Timetable (RMAW05)

Matrix Ensembles and L-Functions

Monday 12th July 2004 to Friday 16th July 2004

Monday 12th July 2004
11:00 to 11:45 C Hughes ([AIM])
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.

12:00 to 12:30 H Montgomery ([Michigan])
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.

14:30 to 15:00 A Ivic ([Belgrade])
On the moments of Hecke series at central points
15:00 to 15:30 M Jutila ([Turku])
The twelth moment of central values of Hecke series
16:00 to 16:45 P Diaconis ([Stanford])
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.

17:00 to 17:30 D Bump ([Stanford])
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).

17:30 to 18:00 I Smolyarenko (University of Cambridge)
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.

18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
Tuesday 13th July 2004
09:30 to 10:20 E Kowalski ([Bordeaux I])
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.

11:00 to 11:45 K Soundararajan ([Michigan])
Extreme values \& moments of L-functions
12:00 to 12:30 A Booker ([Paris-Sud])
Poles of L-functions \& the converse theorem
14:30 to 15:20 P Sarnak ([Princeton])
Perspectives on L functions and spectral theory
16:00 to 16:45 J Keating ([Bristol])
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.

20:30 to 18:00 H Lenstra Jr. ([Leiden])
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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Wednesday 14th July 2004
09:30 to 10:20 D Ulmer ([Arizona])
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.

11:00 to 11:45 N Katz ([Princeton])
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.

12:00 to 12:30 E Duenez ([Texas])
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.

Thursday 15th July 2004
09:30 to 10:20 F Rodriguez-Villegas ([Texas])
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.

11:00 to 11:45 D Farmer ([AIM])
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.

12:00 to 12:30 A Nikeghbali ([Pierre et Marie Curie])
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.

14:30 to 15:20 M Rubinstein ([Waterloo])
Experiments in number theory \& random matrix theory
16:00 to 16:45 S Gonek ([Rochester])
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.

Friday 16th July 2004
09:30 to 10:20 A Gamburd ([Stanford])
Applications of symmetric functions theory to random matrices
11:00 to 11:45 M Zirnbauer ([Koln])
Ratios of random characteristic polynomials from supersymmetry
12:00 to 12:30 C David ([Concordia])
Vanishing of L-functions of elliptic curves over number fields
14:30 to 15:20 D Goldfeld ([Columbia])
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).

16:00 to 16:45 N Snaith ([Bristol])
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.

17:00 to 17:30 A Perelli ([Genova])
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.

17:30 to 17:50 G Molteni ([Milan])
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 The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons