No Title

Abstracts

ALAN BAKER (14h15 on wednesday)
Future prospects in Diophantine Geometry and Transcendence Theory

The study of integer points on algebraic curves lies at the heart of mathematics and many great mathematicians (Fermat, Euler, Gauss, Hilbert, ...) have contributed to the field. It is only in recent times, however, that a general effective method, based on transcendence theory, has been successfully developed in this context. The lecture will survey the current state of the science with particular reference to the theory of logarithmic forms and the abc-conjecture.

VITALY BERGELSON (9h35 on wednesday)
Many facets of the Van der Corput difference trick.

The talk will concentrate on various ramifications and applications of the so called van der Corput difference trick. The connections with harmonic analysis, number theory, combinatorics and ergodic theory will be discussed and some open problems formulated. Some of the results and problems to be mentioned are stemming out of the work of Professor Mendès France and his collaborators.

MARIE-JOSÉ BERTIN (9h00 on monday)
Mesure de Mahler, dilogarithme et equation de Picard-Fuchs.

Liée au problème de Lehmer (1933), introduite par Mahler (1962) pour mesurer la taille de différents facteurs d'un polynôme, au coeur de récentes recherches de Deninger (1997), Boyd (1998) et Rodriguez-Villegas (1999), la mesure de Mahler de polynômes de plusieurs variables permet en outre d'obtenir des formules intégrales explicites de dilogarithmes ainsi que des solutions au voisinage des singularités de certaines équations différentielles de type Picard-Fuchs.

ANNE BERTRAND (16h30 on monday)
Une généralisation des Théorèmes de Pisot-Salem aux nombres de Perron.

Everybody knows that the set of Pisot Numbers is closed. Pisot numbers are real numbers $\theta>1$ that are algebraic numbers whose all conjugates $\theta_i$ other than $\theta$ verify $\vert\theta_i\vert<1$ . We prove the following result : fix an integer k and a number R ; take a sequence $\alpha_n$ of algebraic numbers with limit $\alpha$ and suppose that for every i:
$\bullet$ the conjugates of $\alpha_i$ are all in the disk $\vert z\vert\leqslant R$ .
$\bullet$ $\alpha_i$ has at most k conjugates of modulus $\geqslant 1$ , the other being <1.
Then $\alpha$ is an algebraic integer with at most k conjugates of modulus $\geqslant 1$ and all conjugates in the disk $\vert z\vert\leqslant R$ . Another way to saying the same thing is the following: we say that an algebraic integer $\theta$ dominates its conjugates if for all conjugate $\theta_i$ of $\theta$ $\theta\geqslant \theta_i$ ( $\theta$ is called a Perron-Frobenius number). Let M_k be the set of algebraic numbers that dominate their conjugates and have at most k conjugates of modulus $\geqslant 1$ (M₁ is the set of Pisot numbers, plus 1). Then M_k is closed.

PAULA COHEN (11h15 on thursday)
Aspects arithmétiques de groupes non-arithmétiques.

Dans cet exposé nous étudions comment les groupes fuchsiens non-arithmétiques peuvent figurer dans l'étude de la géométrie et de l'arithmétique d'objets modulaires.

HÉDI DABOUSSI (10h40 on wednesday)
Majoration élémentaire de sommes d'exponentielles.

On majore élémentairement la valeur moyenne

$\begin{displaymath}\frac{1}{x}\sum_{n\leq x}\Lambda(n)e(n\alpha)\end{displaymath}$

par une quantité tendant vers zéro pour les $\alpha$ irrationnels. Ceci améliore un résultat de I.M. Vinogradov d'un facteur $(\log x)^{\varepsilon}.$

MICHEL DEKKING (17h05 on wednesday)
Uniform distribution modulo one: a viewpoint from a tree.

Any sequence of (distinct) real numbers determines a binary tree by storing the numbers consecutively at the nodes according to a right-left algorithm -- or equivalently, by sorting the numbers according to the algorithm Quicksort. Can we see from this tree that the sequence is uniformly distributed modulo 1?

BERNARD DERRIDA (14h50 on tuesday)
Transitions de phase dans des automates stochastiques à une dimension.

Le calcul de l'état stationnaire de certains automates stochastiques peut s'écrire sous une forme algébrique: le poids des configurations du système s'écrit sous la forme d'un produit de matrices dont les règles de commutation permettent de satisfaire la stationarité. En utilisant cette approche, on peut résoudre le problème de l'exclusion asymétrique (ASEP) à une dimension, qui représente un modèle élémentaire de trafic ou de queue et décrire les transitions de phase entre phase embouteillée et phase fluide.

CHRISTOPHE DOCHE (11h15 on monday)
Real roots of random polynomials.

By a result of Erdos and Offord, it is known that the number $\rho$ of real roots of a polynomial P with degree n and $\pm 1$ coefficients verifies $\rho(P)\sim 2/\pi\log n$ . Michel Mendès France asked about the links between the randomness of $\alpha\in\{-1,1\}^\mathbb{N}$ and the number of real roots of the successive polynomials $f_{\alpha,n}(X)=\sum_{i=0}^n \alpha_i X^i$ . The sequence $\alpha$ is said to be mim-random if

$\begin{displaymath}\frac{\rho(f_{\alpha,n})\pi}{2\log n}\mathrel{\mathop{\longrightarrow}_{n \to \infty}} 1.\end{displaymath}$

The point is to exhibit a mim-random sequence, even though we conjecture that almost all sequences $\alpha\in\{-1,1\}^\mathbb{N}$ behave like that. Numerical evidences as well as more theoretical results concerning the Thue-Morse sequence are discussed.

MAURICE DODSON (10h40 on thursday)
Sets of incongruent residues, Shannon's sampling theorem and Plancherel's theorem.

Sets of incongruent residues underlie a trigonometric polynomial analogue of a general form of Shannon's sampling theorem, a cornerstone of information theory. The theorem can be regarded as a convolution over a cyclic subgroup of the real line and is equivalent to a Plancherel type theorem and to the bandwidth limitation in Shannon's theorem.

ÉTIENNE FOUVRY (14h15 on monday)
Équirepartition des valeurs d'un polynôme et géométrie algébrique.

Par des méthodes de majoration de sommes trigonométriques dues à N. Katz et G. Laumon, nous nous intéressons à l'équirepartition modulo 1, des valeurs de $P(x_1,\dots, x_n)/p$ , pour P polynôme fixé de $\mathbb{Z} [X_1,\dots,X_n]$ , $x_1,\dots,x_n$ entiers et p nombre premier, tendant vers l'infini.

JEAN-PIERRE KAHANE (9h00 on wednesday)
Triangles.

Il s'agit de la formule de Girard et de commentaires sur deux de ses démonstrations.

DAVID MASSON (16h30 on wednesday)
Fonctions zonales du groupe symétrique.

Soit G=S_n le groupe symétrique, $s=(s_{0},s_{1},\dots,s_{q-1})$ une partition de n, $H=S_{s_{0}}\times S_{s_{1}}\times \dots \times S_{s_{q-1}}$ le sous groupe de Young associé à s. G/H n'est pas en général un espace symétrique mais on peut quand même définir les fonctions zonales. Dans cet exposé nous présenterons un algorithme de calcul de ces fonctions et nous essaierons de montrer leur intérêt en combinatoire, notamment pour la caractérisation de designs.

WLADYSLAW NARKIEWICZ (15h25 on monday)
Results and problems in polynomial mappings.

A survey will be given of problems and recent results concerning finite orbits of polynomial mappings in various rings.

ANDREW POLLINGTON (9h35 on tuesday)
On Littlewood's conjecture in Diophantine approximation.

This is joint work with Sanju Velani of Queen Mary College, London. We show that the Littlewood conjecture holds for a large class of pairs (x,y) where both x and y are badly approximable numbers. In particular we show that for such pairs $q\Vert qx\Vert \Vert qy\Vert <1/\log q$ infinitely often.

ALF VAN DER POORTEN (14h50 on wednesday)
Composition of Quadratic Forms, of Ideals, and of Continued Fractions.

I tell the story of the magic matrix of Daniel Shanks (and of Gauss) explaining composition of quadratic forms from very first principles. This is a somewhat unusual example where the best explanation and proof of the basic facts also provides the optimal algorithm for computational purposes.

MARTINE QUEFFÉLEC (15h25 on wednesday)
On some transcendance problems.

We investigate problems and results on the transcendence of some real numbers, defined by their expansion : adic-expansion or continued fraction expansion.

GÉRARD RAUZY (9h35 on thursday)
Relations interdites.

On construit progressivement une suite d'entiers ou certaines relations sont interdites. Exemples typiques : x+y=z ou x+y=2z. L'algorithme glouton est-il le meilleur ? Formulations en termes d'automates cellulaires.

IMRE Z. RUZSA (11h15 on tuesday)
A theorem of Hovanski.

Let A be a finite set in a commutative semigroup. A theorem of Hovanski asserts that the function f(n)=|Aⁿ| is a polynomial for large values of n. We present a simple direct proof of this fact and a slight generalization. (Joint work with M. B. Nathanson)

BAHMAN SAFFARI (17h05 on monday)
Sur la conjecture de D. J. Newman concernant la norme L₁ des polynômes.

Cette conjecture de Newman, qui remonte à il y a environ 50 ans, est en effet un sujet assez <<bordelais>> : En 1996, à l'invitation de Michel Mendès France, je fis un exposé à Bordeaux sur cette conjecture, ce qui entraîna une amélioration des résultats par Habsieger. Un nouveau progrès est maintenant réalisé (juin 2000), que je souhaite exposer.

ANDRZEJ SCHINZEL (10h40 on monday)
Wolfgang Schmidt's problem on polynomials.

Let us denote by A(m,n,K), the supremum of the number of non-zero coefficients of (f,g), where f and g run through all univariate polynomials over the field K with m and n non-zero coefficients respectively. It will be shown that if $1<m\leqslant n$ then $A(m,n,K)= +\infty$ , except for (m,n)=(2,2) and (2,3), $\mathop{\mathrm {char}}\nolimits K=0$ , and possibly for (m,n)=(3,3), $\mathop{\mathrm {char}}\nolimits K=0.$

JEFFREY SHALLIT (14h15 on tuesday)
k-regular Sequences.

A sequence $(a(n))_{n \geq 0}$ is k-regular if there exist a finite number of sequences $(b_i (n))_{n \geq 0}$ , $1 \leq i \leq N$ , such that every subsequence of the form $(a(k^e n + c))_{n \geq 0}$ , with $e \geqslant 0$ and $0 \leqslant c < k^e$ can be written as a $\mathbb{Z}$ -linear combination of the b_i. Thus k-regular sequences are a generalization of the k-automatic sequences popularized by Mendès France and others.

CHRIS SMYTH (9h35 on monday)
Zero-mean cosine polynomials and Dirichlet's Theorem.

Take a zero-mean cosine polynomial with non-negative coefficients. For how long can it avoid being negative? The extremal polynomials answering this question can be used for diophantine approximation.

GÉRALD TENENBAUM (11h15 on wednesday)
Séries trigonométriques à coefficients arithmétiques (en collaboration avec Régis de La Bretèche).

Posons $B(\vartheta):=\vartheta-[\vartheta]-\frac12$ si $\vartheta\in\mathbb{R}\setminus \mathbb{Z}$ et $B(\vartheta):=0$ si $\vartheta\in\mathbb{Z}$ . Davenport a montré en 1937 que la relation

$\begin{displaymath}\sin 2\pi\vartheta=-\pi \sum_{n=1}^\infty \frac{\mu(n)}{n}B(n\vartheta)\end{displaymath}$

a lieu pour tout nombre réel $\vartheta$ , la convergence étant uniforme en $\vartheta$ . Nous étudions, à l'aide d'une méthode nouvelle reposant sur un procédé de sommation arithmétique, les conditions de validité de diverses extensions de cette identité. Nous obtenons par exemple que, si l'on désigne par $\tau(m)$ le nombre des diviseurs d'un entier m, alors on a

$\begin{displaymath}\sum_{m=1}^\infty \frac{\tau(m)}{\pi m}\sin 2\pi m\vartheta+ \sum_{n=1}^\infty \frac{B(n\vartheta)}{n}=0\end{displaymath}$

si et seulement si, $\vartheta$ est rationnel ou la série

$\begin{displaymath}\sum_{m=1}^\infty (-1)^m\frac{\log q_{m+1}}{\log q_m}\end{displaymath}$

converge lorsque ${q_m}_{m=1}^\infty$ est la suite des dénominateurs des réduites de $\vartheta$ .

ZHI-YING WEN (10h40 on tuesday)
Some properties of invertible substitution.

Some new results about invertible substitutions will be presented. In particular a sufficient and necessary condition such that two invertible substitutions are locally isomorphic.

JIA-YAN YAO (17h05 on tuesday)
Opacities of finite automata.

A finite automaton is a machine which transforms any sequence over an alphabet into a sequence over another alphabet. Its opacity, introduced by Michel Mendès France in 1991, measures the distortion between the input and the output. In this talk, we shall discuss the general properties of the opacity function. In particular, we shall give an algorithm to compute the opacity of a given automaton and characterize transparent automata whose opacities are zero and opaque automata whose opacities are maximal.