We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory.

2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)

According to the Church-Turing Thesis a number function is computable by the mathematically defined Turing machine if and only if it is computable by a physical machine. In 1983 Pour-El and Richards defined a three-dimensional wave $u(t,x)$ such that the amplitude $u(0,x)$ at time 0 is computable and the amplitude $u(1,x)$ at time 1 is continuous but not computable. Therefore, there might be some kind of wave computer beating the Turing machine. By applying the framework of Type 2 Theory of Effectivity (TTE), in this paper we analyze computability of wave propagation. In particular, we prove that the wave propagator is computable on continuously differentiable waves, where one derivative is lost, and on waves from Sobolev spaces. Finally, we explain why the Pour-El-Richards result probably does not help to design a wave computer which beats the Turing machine.

2000 Mathematical Subject Classification: 03D80, 03F60, 35L05, 68Q05.

Let ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ be the Hilbert scheme of closed 1-dimensional subschemes of degree 6 and arithmetic genus 4 in $\mathbb{P}^3$. Let $H$ be the component of ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ whose generic point corresponds to a canonical curve, that is, a complete intersection of a quadric and a cubic surface in $\mathbb{P}^3$. Let $F$ be the vector space of linear forms in the variables $z_1, z_2, z_3, z_4$. Denote by $F_d$ the vector space of homogeneous forms of degree $d$. Set $X = \{(f_2,f_3)\}$ where $f_2 \in \mathbb{P}(F_2)$ is a quadric surface, and $f_3 \in \mathbb{P}(F_3/f_2 \cdot F)$ is a cubic modulo $f_2$. We have a rational map, $\sigma : X \cdots \rightarrow H$ defined by $(f_2,f_3) \mapsto f_2 \cap f_3$. It fails to be regular along the locus where $f_2$ and $f_3$ acquire a common linear component. Our main result gives an explicit resolution of the indeterminacies of $\sigma$ as well as of the singularities of $H$.

2000 Mathematical Subject Classification: 14C05, 14N05, 14N10, 14N15.

A first aim of this paper is to answer, in the case of real ruled surfaces $X_l$ of base $\mathbb{C}P^1$, with $l \geq 2$, a question of V. A. Rokhlin: is it true that the equivariant isotopy class does not suffice to distinguish the connected components of the space of smooth real algebraic curves of $X_l$? A second aim is to prove that there exist in these surfaces some real schemes realized by real flexible curves but not by smooth real algebraic curves. These two results of real algebraic geometry are deduced from the following comparison theorem: when $m = l + 2k$, with $k > 0$, the discriminants of the surface $X_m$ are deduced from those of the surface $X_l$ via weighted homotheties. All these results are obtained from a study of a deformation of ruled surfaces.

The paper is written in French.

2000 Mathematical Subject Classification: 14H10, 14J26, 14P25.

We study finitely generated modules $M$ over a ring $R$, noetherian on both sides. If $M$ has finite Gorenstein dimension $\mbox{G-dim}_RM$ in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functors ${\rm Ext}^n_R(M,\ )$. Relative cohomology functors ${\rm Ext}^n_{\mathcal G}(M,\ )$ are defined for all non-negative integers $n$; they treat the modules of Gorenstein dimension $0$ as projectives and vanish for $n > \mbox{G-dim}_RM$. Tate cohomology functors $\widehat{\rm Ext}^n_R(M,\ )$ are defined for all integers $n$; all groups $\widehat{\rm Ext}^n_R(M,N)$ vanish if $M$ or $N$ has finite projective dimension. Comparison morphisms $\varepsilon_{\mathcal G}^n \colon {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ )$ and $\varepsilon_R^n \colon {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ )$ link these functors. We give a self-contained treatment of modules of finite G-dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence $0 \to {\rm Ext}^1_{\mathcal G}(M,\ ) \to \cdots \to {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ ) \to {\rm Ext}^{n+1}_{\mathcal G}(M,\ ) \to \cdots$. We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings.

2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.

We extend the range of observed behaviour among length functions of optimal asynchronously automatic structures. We do so by means of a construction that yields asynchronously automatic groups with finite aspherical presentations where the Dehn function of the group is polynomial of arbitrary degree. Many of these groups can be embedded in the automorphism group of a free group. Moreover, the fact that the groups have aspherical presentations makes them useful tools in the search to determine the spectrum of exponents for second order Dehn functions. We contribute to this search by giving the first exact calculations of groups with quadratic and superquadratic exponents.

2000 Mathematical Subject Classification:20F06, 20F65, 20F69.

We define the scenery flow space at a point z in the Julia set J of a hyperbolic rational map $T : \mathbb{C} \to \mathbb{C}$ with degree at least 2, and more generally for T a conformal mixing repellor.

We prove that, for hyperbolic rational maps, except for a few exceptional cases listed below, the scenery flow is ergodic. We also prove ergodicity for almost all conformal mixing repellors; here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real-analytic curves, and furthermore that for the collection of such maps based on a fixed open set U, the ergodic cases form a dense open subset of that collection. Scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the Hausdorff measure $H^d$, where d is the dimension of J, and that the flow has a unique measure of maximal entropy.

For all conformal mixing repellors, the flow is loosely Bernoulli and has topological entropy at most d. Moreover the flow at almost every point is the same up to a rotation, and so as a corollary, one has an analogue of the Lebesgue density theorem for the fractal set, giving a different proof of a theorem of Falconer.

2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.

Bohr's inequality says that if $f(z) = \sum^{\infty}_{n = 0} a_n z^n$ is a bounded analytic function on the closed unit disc, then $\sum^{\infty}_{n = 0} \lvert a_n\rvert r^n \leq \Vert f\Vert_{\infty}$ for $0 \leq r \leq 1/3$ and that $1/3$ is sharp. In this paper we give an operator-theoretic proof of Bohr's inequality that is based on von Neumann's inequality. Since our proof is operator-theoretic, our methods extend to several complex variables and to non-commutative situations.

We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher-dimensional domains, for the non-commutative disc algebra ${\mathcal A}_n$, and for the reduced (respectively full) group C*-algebra of the free group on $n$ generators.

We also include an application to Banach algebras. We prove that every Banach algebra has an equivalent norm in which it satisfies a non-unital version of von Neumann's inequality.

2000 Mathematical Subject Classification: 47A20, 47A56.