Let $\cal V$ be a complete connected hyperbolic 3-manifold of finite volume, with Liouville measure $m$, geodesic flow $\Gamma_t$ and Brownian motion $Z_t$. Let $\omega$ be a smooth 1-form, closed in the cusps of $\cal V$. Then the limit laws as $t \to \infty$ of $(t\log t)^{-1/2}\int_0^t\omega(\Gamma_s)$ under $m$ and of $(t\log t)^{-1/2}\int_0^t\omega(Z_s)$ are calculated, and seen to be Gaussian, and equal. The geodesic flow case is studied via the Brownian case.

1991 Mathematics Subject Classification: 60J65, 58F17, 51M10.

Let $K/k$ be a finite unramified Galois extension of number fields with Galois group $G$. This determines two homomorphisms from the ideal class group $\mathrm{Cl}_k$ of $k$: the capitulation map $\mathrm{Cl}_k \to \mathrm{Cl}_K$ and the Artin map $\mathrm{Cl}_k \twoheadrightarrow G^{\mathrm{ab}}$ onto the abelianization $G^{\mathrm{ab}}$ of $G$. We call (ker (capitulation), $\mathrm{Cl}_k$, Artin) the capitulation triple of $K / k$.

Artin's transition to group theory shows that any triple $(X, Y, \zeta)$ which arises in this way satisfies the group-theoretic property of being a transfer triple for $G$, defined as follows: there exist a group extension $A \rightarrowtail H \twoheadrightarrow G$ with $A$ finite abelian and an isomorphism $\eta : Y \stackrel{\sim}{\to} H^{\mathrm{ab}}$ such that $\eta (X)$ is the kernel of the transfer homomorphism $H^{\mathrm{ab}}\to A$, and $\zeta$ is the composite of $\eta$ with $H^{\mathrm{ab}}\to G^{\mathrm{ab}}$.

When $G$ is abelian, we show that a triple $(X, Y, \zeta)$ is a transfer triple for $G$ if and only if $|G| X = 0$ and $|G|$ divides $|X|$. Whether all transfer triples for $G$ can be realized arithmetically remains an unsolved problem.

Let $G = SL_n(F)$ where $F$ is a non-Archimedean local field. This paper concerns the smooth (complex) representation theory of $G$, specifically, the construction of types in $G$, in the sense of Bushnell and Kutzko. The main result describes a type for each non-supercuspidal component of the Bernstein decomposition of $G$. If this is combined with earlier work of Bushnell and Kutzko, it follows that $G$ admits a complete set of types.

2000 Mathematical Subject Classification: 22E50.

We relate the brace products of a fibration with section to the differentials in its Serre spectral sequence. In the particular case of free loop fibrations, we establish a link between these differentials and Browder operations in the fiber. Applications and several calculations (for the particular cases of spheres and wedges of spheres) are given.

2000 Mathematical Subject Classification: 55P35, 55Q15, 55R05, 55R20.

Thom--Boardman strata $\Sigma^I$ are fundamental tools in studying singularities of maps. The Zariski closures of the strata $\Sigma^I$ are components of the set of zeros of the ideals $\Delta^I$ defined by B. Morin using iterated jacobian extensions in his paper `Calcul jacobien' ({\em Ann. Sci. \'Ecole Norm. Sup.} 8 (1975) 1--98). In this paper, we consider the question of when the Morin ideals $\Delta^I$ define Cohen--Macaulay spaces. We determine all $I=(i_1,...,i_k)$ such that $\Delta^I$ defines a Cohen--Macaulay space alongthe $\Sigma^{i_1}$ stratum. 1991 Mathematics Subject Classification: 13D25, 14B05, 14M12, 58C25.

This paper gives one-term componentwise asymptotics for the $M$ and spectral matrices of a self-adjoint realisation of an even-order ordinary differential expression. The underlying interval is assumed to have at least one regular endpoint, and the boundary conditions are supposed to be separated. Furthermore, the weight function and the reciprocal of the highest-order coefficient are supposed to be of regular variation at the regular endpoint, in the sense of Bingham, Goldie and Teugels.

1991 Mathematics Subject Classification: 34B24, 34E05.

Let $f$, $g$ : $({\Bbb R}^n,0)\rightarrow ({\Bbb R}^p,0)$ be two $C^\infty$ map-germs. Then $f$ and $g$ are $C^0$-equivalent if there exist homeomorphism-germs $h$ and $l$ of $({\Bbb R}^n,0)$ and $({\Bbb R}^p,0)$ respectively such that $g=l\circ f\circ h^{-1}$. Let $k$ be a positive integer. A germ $f$ is $k$-$C^0$-determined if every germ $g$ with $j^k g(0)=j^k f(0)$ is $C^0$-equivalent to $f$. Moreover, we say that $f$ is finitely topologically determined if $f$ is $k$-$C^0$-determined for some finite $k$. We prove a theorem giving a sufficient condition for a germ to be finitely topologically determined. We explain this condition below.

Let $N$ and $P$ be two $C^\infty$ manifolds. Consider the jet bundle $J^k(N,P)$ with fiber $J^k(n,p)$. Let $z\in J^k(n,p)$ and let $f$ be such that $z= j^kf(0)$. Define \[ \chi(f)=\dim_{\Bbb R} \frac{\theta (f)} {tf(\theta (n))+f^{\ast}(m_p)\theta(f)}. \] Whether $\chi(f)(k$ depends only on $z$, not on $f$ . We can therefore define the set $W^k= W^k(n,p)=\{z\in J^k(n,p)\vert \chi(f)\ge k$ for some representative $f$ of $z$\}. Let $W^k(N,P)$ be the subbundle of $J^k(N,P)$ with fiber $W^k(n,p)$. Mather has constructed a finite Whitney (b)-regular stratification ${\mathcal{S}}^k(n,p)$ of $J^k(n,p)-W^k(n,p)$ such that all strata are semialgebraic and $\mathcal K$-invariant, having the property that if ${\mathcal S}^k(N,P)$ denotes the corresponding stratification of $J^k(N,P)-W^k(N,P)$ and $f\in C^\infty (N,P)$ is a $C^\infty$ map such that $j^k f$ is multitransverse to ${\mathcal S}^k(N,P)$, $j^k f(N)\cap W^k(N,P)=\emptyset$ and $N$ is compact (or $f$ is proper), then $f$ is topologically stable.

For a map-germ $f:({\Bbb R}^n,0)\rightarrow ({\Bbb R}^p,0)$, we define a certain {\L}ojasiewicz inequality. The inequality implies that there exists a representative $f : U\rightarrow{\Bbb R}^p$ such that $j^kf(U-\{0\})\cap W^k({\Bbb R}^n,{\Bbb R}^p)=\emptyset$ and such that $j^kf$ is multitransverse to ${\mathcal S}^k({\Bbb R}^n,{\Bbb R}^p)$ at any finite set of points $S\subset U-\{0\}$. Moreover, the inequality controls the rate $j^k f$ becomes non-transverse as we approach 0. We show that if $f$ satisfies this inequality, then $f$ is finitely topologically determined.

1991 Mathematics Subject Classification: 58C27.

We compute the Euler characteristics of the individual connected components of the intersection of two opposed big cells in the real flag variety of type G2, verifying a conjecture of Rietsch [6].

2000 Mathematical Subject Classification: primary 14M15; secondary 20G20.

We exploit duality considerations in the study of singular combinatorial 2-discs (diagrams) and are led to the following innovations concerning the geometry of the word problem for finite presentations of groups. We define a filling function called gallery length that measures the diameter of the 1-skeleton of the dual of diagrams; we show it to be a group invariant and we give upper bounds on the gallery length of combable groups. We use gallery length to give a new proof of the Double Exponential Theorem. Also we give geometric inequalities relating gallery length to the space-complexity filling function known as filling length.

{\frenchspacing Nous \'etudions l'homotopie d'une vari\'et\'e quasi-projective dans un espace projectif complexe selon la m\'ethode de Lefschetz, c'est-\`a-dire en consid\'erant ses sections par les hyperplans d'un pinceau (tomographie). En particulier, nous aboutissons \`a un th\'eor\`eme du type de Lefschetz qui g\'en\'eralise dans une certaine direction les meilleurs r\'esultats connus dus \`a Hamm, L\^e, Goresky et MacPherson. Ce th\'eor\`eme est d\'emontr\'e par r\'ecurrence sur la dimension de l'espace projectif ambiant \`a partir d'un th\'eor\`eme sur les pinceaux d'axe g\'en\'erique qui constitue le r\'esultat principal de l'article. Ce dernier compare la topologie de la vari\'et\'e \`a celle de sa section par un hyperplan g\'en\'erique du pinceau sur la base des comparaisons (section hyperplane g\'en\'erique -- section par l'axe du pinceau) et (sections hyperplanes exceptionnelles -- section par l'axe); l'incidence des singularit\'es est mesur\'ee par un invariant appel\'e `profondeur homotopique rectifi\'ee globale' (analogue global de la notion de profondeur homotopique rectifi\'ee de Grothendieck).} \vspace{6mm} \noindent We study the homotopy of a quasi-projective variety in a complex projective space following Lefschetz's method, that is, by considering its sections by the hyperplanes of a pencil (tomography). Specifically, we obtain a theorem of Lefschetz type which generalizes in a certain direction the best-known results due to Hamm, L\^e, Goresky and MacPherson. This theorem is proved by induction on the dimension of the ambient projective space with the help of a theorem on pencils with generic axis which is the main result of the paper. The latter compares the topology of the variety with that of its section by a generic hyperplane of the pencil, on the basis of the following comparisons: section by a generic hyperplane with section by the axis of the pencil; and sections by the exceptional hyperplanes with section by the axis. The effect of the singularities is measured by an invariant called `global rectified homotopical depth' (a global analogue of the notion of rectified homotopical depth of Grothendieck). E-mail: [email protected] 2000 Mathematics Subject Classification: 32S50, 14F35, 14F17.

Smooth projective varieties with small invariants have received renewed interest in recent years, primarily due to the fine study of the adjunction mapping. Now, through the effort of several mathematicians, a complete classification of smooth surfaces in ${\Bbb P}^4$ has been worked out up to degree $10$, and a partial one is available in degree $11$. On the other side, recently Ellingsrud and Peskine have proved Hartshorne's conjecture that there are only finitely many families of smooth surfaces in ${\Bbb P}^4$, not of general type. It is believed that the degree of the smooth, non-general type surfaces in ${\Bbb P}^4$ should be less than or equal to $15$.

The aim of this paper is to provide a series of examples of smooth surfaces in ${\Bbb P}^4$, not of general type, in degrees varying from $12$ up to $14$, and to describe their geometry. By using mainly syzygies and liaison techniques, we construct the following families of surfaces: \begin{enumerate} \item[] minimal proper elliptic surfaces of degree $12$ and sectional genus $\pi=13$; \item[] two types of non-minimal proper elliptic surfaces of degree $12$ and sectional genus $\pi=14$; \item[] non-minimal $K3$ surfaces of degree $13$ and sectional genus $16$; and \item[] non-minimal $K3$ surfaces of degree $14$ and sectional genus $19$.

1991 Mathematics Subject Classification: 14M07, 14J25, 14J26, 14J28, 14C05.

Let $\cal D $ denote the collection of dyadic intervals in the unit interval. Let $\tau$ be a rearrangement of the dyadic intervals. We study the induced operator

$$ Th_I = h_{\tau(I)}$$

where $h_I$ is the $L^{\infty}$ normalized Haar function. We find geometric conditions on $\tau$ which are necessary and sufficient for $T$ to be bounded on $BMO$. We also characterize the rearrangements of the $L^p$ normalized Haar system in $L^p.$

1991 Mathematics Subject Classification: 42C20, 43A17, 47B38.

We study the Hardy inequality for a class of manifolds equipped with a singular metric that becomes infinite on the boundary; in particular, we study the best constant possible. One application of this constant has been to a certain type of domain perturbation. This technique is useful when the domain has an irregular boundary as is the case here. However, our first result shows that the class of manifolds considered has a Hardy constant that lies outside the range permitted by existing theorems. Yet we are still able to prove theorems which give information about the domain perturbation problem and, moreover, we set up a specific example which can be used to show that our results are the best possible.

2000 Mathematical Subject Classification: 35P99, 47A75,47B25,58J99.

We apply set-theoretical ideas to an iteration problem of dynamical systems. Among other results, we prove that these iterations never stabilise later than the first uncountable ordinal; for every countable ordinal we give examples in Baire space and in Cantor space of an iteration that stabilises exactly at that ordinal; we give an example of an iteration with recursive data which stabilises exactly at the first non-recursive ordinal; and we find new examples of complete analytic sets simply definable from concepts of recurrence.

2000 Mathematics Subject Classification: primary 03E15, 37B20, 54H05; secondary 37B10, 37E15.

Suppose $\mathcal{O}$ is either the ring of integers of a number field, the ring of integers of a $p$-adic local field, or a field of characteristic $0$. Let $\mathcal{X}$ be a regular projective scheme which is flat and equidimensional over $\mathcal{O}$ of relative dimension $d$. Suppose $G$ is a finite group acting tamely on $\mathcal{X}$. Define ${\rm HCl}(\mathcal{O} G)$ to be the Hermitian class group of $\mathcal{O} G$. Using the duality pairings on the de Rham cohomology groups $H^*(X, \Omega^\bullet_{X / F})$ of the fiber $X$ of $\mathcal{X}$ over $F = {\rm Frac}(\mathcal{O})$, we define a canonical invariant $\chi_H(\mathcal{X}, G)$ in ${\rm HCl}(\mathcal{O} G)$ . When $d = 1$ and $\mathcal{O}$ is either $\mathbb{Z}$, $\mathbb{Z}_p$ or $\mathbb{R}$, we determine the image of $\chi_H(\mathcal{X}, G)$ in the adelic Hermitian classgroup ${\rm Ad\,HCl}(\mathbb{Z} G)$ by means of $\epsilon$-constants. We also show that in this case, the image in ${\rm Ad\,HCl}(\mathbb{Z} G)$ of a closely related Hermitian Euler characteristic $\chi_{H}(\mathcal{X}, G)(0)$ both determines and is determined by the $\epsilon_0$-constants of the symplectic representations of $G$.

Let $\ast$ denote convolution and let $\delta_x$ denote the Dirac measure at a point $x$. A function in $L^2({\mathbb{R}})$ is called a difference of order $1$ if it is of the form $ g- \delta_{x} \ast g$ for some $x \in {\mathbb{R}}$ and $g \in L^2 ({\mathbb{R}})$. Also, a difference of order $2$ is a function of the form $ g- 2^{-1}( \delta_{x} \ast g + \delta_{-x} \ast g)$ for some $x \in {\mathbb{R}}$ and $g \in L^2 ({\mathbb{R}})$. In fact, the concept of a ‘difference of order $s$’ may be defined in a similar manner for each $s > 0$. If $\widehat{f}$ denotes the Fourier transform of $f$, it is known that a function $f$ in $L^2 ({\mathbb{R}})$ is a finite sum of differences of order $s$ if and only if $\int_{-\infty}^{\infty} |\widehat {f}(x)|^2 |x|^{-2s} dx < \infty$, and the vector space of all such functions is denoted by ${\mathcal{D}}_{s} (L^2 ({\mathbb{R}}) )$. Every function in ${\mathcal{D}}_{s} (L^2 ({\mathbb{R}}) )$ is a sum of ${\mathrm{int}} (2s) + 1$ differences of order $s$, where ${\mathrm{int}} (t)$ denotes the integer part of $t$. Thus, every function in ${\mathcal{D}}_{1} (L^2 ({\mathbb{R}}) )$ is a sum of three first order differences, but it was proved in 1994 that there is a function in ${\mathcal{D}}_1 (L^2 ({\mathbb{R}}) )$ which is never the sum of two first order differences. This complemented, for the group ${\mathbb{R}}$, the corresponding result for first order differences obtained by Meisters and Schmidt in 1972 for the circle group. The results show that there is a function in $L^2 ({\mathbb{R}})$ such that, for each $s \ge 1/2$, this function is a sum of ${\mathrm{int}} (2s) + 1$ differences of order $s$ but it is never the sum of ${\mathrm{int}} (2s)$ differences of order $s$. The proof depends upon extending to higher dimensions the following result in two dimensions obtained by Schmidt in 1972 in connection with Heilbronn's problem: if $x_1, \ldots, x_n$ are points in the unit square,

$$\sum_{1 \le i < j \le n} |x_i - x_j|^{-2} \ge 200^{-1} n^2 \ln n.$$

Following on from the work of Meisters and Schmidt, this work further develops a connection between certain estimates in combinatorial geometry and some questions of sharpness in harmonic analysis.

Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let $f\colon Y \to X$ be a map of K-affinoid varieties. In this paper we study the analytic structure of the image $f(Y) \subset X$; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the $\mathbf D$-semianalytic sets, where $\mathbf D$ is the truncated division function first introduced by Denef and van den Dries. This result is most conveniently stated as a Quantifier Elimination result for the valuation ring R in an analytic expansion of the language of valued rings. To prove this we establish a Flattening Theorem for affinoid varieties in the style of Hironaka, which allows a reduction to the study of subanalytic sets arising from flat maps, that is, we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of a flat map is then dealt with by a small extension of a result of Raynaud and Gruson showing that the image of a flat map of affinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in the zero characteristic case, a Uniformization Theorem for subanalytic sets: a subanalytic set can be rendered semianalytic using only finitely many local blowing ups with smooth centres. As a corollary we obtain the fact that any subanalytic set in the plane R2 is semianalytic. 2000 Mathematical Subject Classification: 32P05, 32B20, 13C11, 12J25, 03C10.

1991 Mathematics Subject Classification: 14J10, 32J15.

A semi-algebra of continuous functions is a cone $A$ of continuous real functions on a compact Hausdorff space $X$ such that $A$ contains the products of its elements. A cone $A$ is said to be of type $n$ if $f\in A$ implies $f^n(1 + f)^{-1} \in A$. Uniformly closed semi-algebras of types 0 and 1 have long been characterized in a manner analogous to the Stone--Weierstrass theorem, but, except for the case when $A$ is generated by a single function, little has been known about type 2. Here, progress is reported on two problems. The first is the characterization of those continuous linear functionals on $C(X)$ that determine semi-algebras of type 2. The second is the determination of the type of the tensor product of two type 1 semi-algebras. 1991 Mathematics Subject Classification: 46J10.

For a dense $G_\delta$-set of parameters, the irrational rotation algebra is shown to contain infinitely many C*-subalgebras satisfying the following properties. Each subalgebra is isomorphic to a direct sum of two matrix algebras of the same (perfect square) dimension; the Fourier transform maps each summand onto the other; the corresponding unit projection is approximately central; the compressions of the canonical generators of the irrational rotation algebra are approximately contained in the subalgebra.