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.

Let $A$ be a commutative, cocommutative Hopf algebra, finitely generated and projective over its base ring $R$. Waterhouse asked whether the image of the class-invariant map, taking each $A$-Galois algebra to the class in ${\rm Pic}(A)$ of its $R$-linear dual, is the group of primitive classes in ${\rm Pic}(A)$. We discuss functorial aspects of this problem, and relate it to Fr\"ohlich's Hom-description of ${\rm Pic}(A)$ in the case that $R$ is a Dedekind domain with field of fractions $K$, and $A$ is an $R$-Hopf order in a separable $K$-Hopf algebra. We then apply this machinery to a certain class of Hopf orders $\mathfrak A$ in the Hopf algebra ${\rm Map}(G,K)$. More precisely, we give a positive answer to Waterhouse's question for $\mathfrak A$ when the dual $\mathfrak B$ of $\mathfrak A$ is one of the Hopf orders in $KG$ constructed by Larson, and a compatibility condition holds between the filtrations of $G$ determined by the various completions of $\mathfrak B$.

1991 Mathematics Subject Classification: 11R33, 16W30

The author investigates the solubility in rationals of equations of the form $$a_0X_0^4 + a_1X_1^4 + a_2X_2^4 + a_3X_3^4 = 0 $$ where $a_0a_1a_2a_3$ is a square, building on the ideas which Colliot-Thène, Skorobogatov and he have developed; see {\em Invent. Math.} 134 (1998) 579--650. He obtains sufficient conditions for solubility, which appear to be related to the absence of a Brauer--Manin obstruction. This represents the first large family of K3 surfaces which almost satisfy the Hasse principle, in the sense that the auxiliary condition which ensures that local solubility everywhere implies global solubility is nearly always satisfied. 1991 Mathematics Subject Classification: 10B10.

If $\mathbb{C}[G] \hookrightarrow \mathbb{C}[H]$ is an extension of Hopf domains of degree $d$, then $H \twoheadrightarrow G$ is an \'etale map. Equivalently, the variety $X_{\mathbb{C}[H]}$ of $d$-dimensional $\mathbb{C}[H]$-modules compatible with the trace map of the extension, is a smooth $\mbox{GL}_d$-variety with quotient $G$. If we replace $\mathbb{C}[H]$ by a non-commutative Hopf algebra $H$, we construct similarly a $\mbox{GL}_d$-variety and quotient map $\pi : X_H \twoheadrightarrow G$. The smooth locus of $H$ over $\mathbb{C}[G]$ is the set of points $g \in G$ such that $X_H$ is smooth along$\pi^{-1}(g)$. We relate this set to the separability locus of $H$ over $\mathbb{C}[G]$ as well as to the (ordinary) smooth locus of the commutative extension $\mathbb{C}[G] \hookrightarrow Z$ where $Z$ is the centre of $H$. In particular, we prove that the smooth locus coincides with the separability locus whenever $H$ is a reflexive Azumaya algebra. This implies that the quantum function algebras $O_{\epsilon}(G)$ and quantised enveloping algebras $U_{\epsilon}(\mathfrak{g})$ are as singular as possible. 1991 Mathematics Subject Classification: 16W30, 16R30.

Let $\mathbf G$ be a connected reductive group defined over a finite field with $q$ elements and let $F : {\mathbf G} \to {\mathbf G}$ be the corresponding Frobenius endomorphism. We prove that the Mackey formula for Lusztig induction and restriction holds in ${\mathbf G}^F$ if all the simple components of ${\mathbf G}$ are of type A. Our result holds for every Frobenius endomorphism (split or non-split) and for every $q$. 1991 Mathematics Subject Classification: 20G05, 20G40.

To study the distribution of pairs of zeros of the Riemann zeta-function, Montgomery introduced the function $$ F(\alpha) = F_T(\alpha) = \left({T\over 2\pi}\log T\right)^{-1} \sum_{0<\gamma,\gamma ' \le T} T^{i\alpha(\gamma -\gamma ')}w(\gamma-\gamma '), $$ where $\alpha$ is real and $T\ge 2$, $\gamma$ and $\gamma '$ denote the imaginary parts of zeros of the Riemann zeta-function, and $w(u) = 4/(4 + u^2)$. Assuming the Riemann Hypothesis, Montgomery proved an asymptotic formula for $F(\alpha)$ when $|\alpha|\le 1$, and made the conjecture that $F(\alpha) = 1 + o(1)$ as $T\to \infty$ for any bounded $\alpha$ with $|\alpha |\ge 1$. In this paper we use an approximation for the prime indicator function together with a new mean value theorem for long Dirichlet polynomials and tails of Dirichlet series to prove that, assuming the Generalized Riemann Hypothesis for all Dirichlet $L$-functions, then for any $\epsilon >0$ we have $$ F(\alpha) \ge {3\over 2} - |\alpha| - \epsilon ,$$ uniformly for $1\le |\alpha| \le \frac32 -2\epsilon $ and all $T \ge T_0(\epsilon)$.

1991 Mathematics Subject Classification: primary 11M26; secondary 11P32.

If $G$ is a projective special linear group $\text{PSL}(3,q)$ with $q \equiv 4 \; \text{or} \; 7 \pmod{9}$, then a Sylow 3-subgroup of $G$ is elementary abelian of order 9. We show that the principal 3-blocks of any two such groups are Morita equivalent. This result and Okuyama's theorem for $\text{PSL}(3,4)$ prove the Broué conjecture for these blocks. 1991 Mathematics Subject Classification: 20C05, 20C20.

The notion of \emph{hyperdecidability} has been introduced by the first author as a tool to prove decidability of semidirect products of pseudovarieties of semigroups. In this paper we consider some stronger notions which lead to improved decidability results allowing us in turn to establish the decidability of some iterated semidirect products. Roughly speaking, the decidability of a semidirect product follows from a mild, commonly verified property of the first factor plus the stronger property for all the other factors. A key role in this study is played by intermediate free semigroups (relatively free objects of expanded type lying between relatively free and relatively free profinite objects). As an application of the main results, the decidability of the Krohn--Rhodes (group) complexity is shown to follow from non-algorithmic abstract properties likely to be satisfied by the pseudovariety of all finite aperiodic semigroups, thereby suggesting a new approach to answer (affirmatively) the question as to whether complexity is decidable.

1991 Mathematics Subject Classification: primary 20M05, 20M07, 20M35; secondary 08B20.

Let $G$ be a simply-connected, semisimple algebraic group over $k$, an algebraically closed field of characteristic zero. Let ${\cal O}_{\epsilon}[G]$ be the quantized function algebra of $G$ at a primitive $\ell$th root of unity $\epsilon$, and let $\overline{{\cal O}_{\epsilon}[G]}$ be the `restricted' quantized function algebra at $\epsilon$, a finite-dimensional $k$-algebra obtained from ${\cal O}_{\epsilon}[G]$ by factoring out a centrally generated ideal. It is known that $\overline{{\cal O}_{\epsilon}[G]}$ is a Hopf algebra. We study the cohomology ring $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$, a graded commutative algebra, and, for any finite-dimensional $\overline{{\cal O}_{\epsilon}[G]}$-module $M$, the $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$-module $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$. We prove that for sufficiently large $\ell$ there is an isomorphism of graded algebras\[ \mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k) \cong k[X_1,\ldots ,X_{2N}],\] where each $X_i$ is homogeneous of degree $2$, and $2N$ equals the number of roots associated to $G$. Moreover we show that in this case $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$is a finitely generated $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$-module. We also show under much less restrictive conditions on $\ell$ that $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$ continues to be a finitely generated graded commutative algebra over which $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$ is a finitely generated module.\vspace{6mm}\noindent 1991 Mathematics Subject Classification: 16W30, 17B37, 17B56.

Let $G$ be a locally compact group. The question of whether ${\cal H}^1(L^1(G),M(G))$, the first Hochschild cohomology group of $L^1(G)$ with coefficients in $M(G)$, is zero was first studied by B. E. Johnson and initiated his development of the theory of amenable Banach algebras. He was able to show that ${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$ whenever $G$ is amenable, a $[SIN]$-group, or a matrix group satisfying certain conditions. No group such that ${\cal H}^1(L^1(G),M(G)) \neq \{ 0 \}$ is known. In this paper, we approach the problem of whether ${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$ from several angles. Using weakly almost periodic functions, we show that ${\cal H}^1(L^1(G),L^1(G))$ is always Hausdorff for unimodular $G$. We also show that for $[IN]$-groups, every derivation $D \colon L^1(G) \to L^1(G)$ is implemented, not necessarily by an element of $M(G)$, but at least by an element of $\mbox{VN}(G)$, the group von Neumann algebra of $G$. This applies, in particular, to the group $G := {\mathbb T}^2 \rtimes \mbox{SL}(2, {\mathbb Z})$, for which it is unknown whether ${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$. Finally, we analyse the structure of derivations on $L^1(G)$; an important r\^ole is played by the closed normal subgroup $N$ of $G$ generated by the elements of $G$ with relatively compact conjugacy classes. We can write an arbitrary derivation $D \colon L^1(G) \to L^1(G)$ as a sum $D = D_N + D_{N^\perp}$, where $D_N$ and $D_{N^\perp}$ can be tackled with different techniques. Under suitable conditions, all satisfied by ${\mathbb T}^2 \rtimes \mbox{SL}(2,{\mathbb Z})$, we can show that $D_N$ is implemented by an element of $\mbox{VN}(G)$ and that $D_{N^\perp}$ is implemented by a measure. 1991 Mathematics Subject Classification: 22D05, 22D25, 43A10, 43A20, 46H25, 46L10, 46M20, 47B47, 47B48.

In this paper we study the rational representation theory of the general linear group $G = \mbox{GL}_n(F)$ over an algebraically closed field $F$ of characteristic $p$. Given $\alpha \in {\Bbb Z} / p{\Bbb Z}$, we define functors $\mbox{Tr}^\alpha$ and $\mbox{Tr}_\alpha$, which, roughly speaking, are given by tensoring with the natural $G$-module $V$ and its dual $V^*$ respectively, and then projecting onto certain blocks determined by the residue $\alpha$. In fact, these functors can be viewed as special cases of Jantzen's translation functors. We prove a number of fundamental properties about these functors and also certain closely related functors that arise in the modular representation theory of the symmetric group.

1991 Mathematics Subject Classification: 20G05, 20C05.

In this paper we study the problem of $L^p$-independence of the spectrum of second-order elliptic operators with measurable coefficients. A new technique is developed to treat the problem in cases when the kernel of the corresponding semigroup does not satisfy upper bounds of Gaussian type and the semigroup itself exists only in a certain interval of the $L^p$-scale. This allows the authors to treat second-order elliptic operators of divergence type with singular coefficients in the main part and singular lower-order terms. A criterion of $L^p$-independence obtained in the paper applies also to certain classes of higher-order elliptic operators. The problem of stability of the essential spectrum is also studied. It is shown, under some mild conditions on the coefficients of the elliptic operators, that its essential spectrum is the same as the one of the Laplacian. 1991 Mathematics Subject Classification: 35P99, 35B25, 35J15.

We consider Hilbert transforms along curves $\Gamma$ on the Heisenberg group. In particular, necessary and sufficient conditions for the $L^2$-boundedness are given for $\Gamma(t)=(t,\gamma(t),0)$ when $\gamma$ is even or odd and convex on ${\Bbb R}^{+}$. 1991 Mathematics Subject Classification: 42B20,42B25.

We give a new non-capacitary characterization of positive Borel measures $\mu$ on ${\Bbb R}^n$ such that the potential space $I_\alpha*L^p$ is imbedded in $L^q(d\mu)$ for $1< q < p<+\infty$, that is, the trace inequality $\|I_\alpha f\|_{L^q(d\mu)} \le C \, \|f\|_{L^p(d x)}$ holds, for Riesz potentials $I_\alpha = (- \Delta)^{-\alpha/2}$. A weak-type trace inequality is also characterized. A non-isotropic version on the unit sphere ${\Bbb S}^n$ is studied, as well as the holomorphic case for Hardy--Sobolev spaces $H_\alpha^p$ in the ball. 1991 Mathematics Subject Classification: primary 31C15, 42B20; secondary 32A35.

In this paper we determine the possible Hilbert functions of a Cohen--Macaulay local ring of dimension $d$ and multiplicity $e$, in the case where the embedding dimension $v$ satisfies $v=e+d-3$ and the Cohen--Macaulay type is less than or equal to $e-3$.

1991 Mathematics Subject Classification: primary 13D40; secondary 13P99.

We find Zariski pairs of sextics with simple singularities having maximal total Milnornumber. We also relate them to the existence of distinct Mordell--Weil groups of extremal elliptic $K3$ surfaces with a fixed set of semistable singular fibres.

1991 Mathematics Subject Classification: 14F45, 14F27, 14F28.

A {\em convex corner} is a compact convex down-set of full dimension in ${\mathbb R}_+^n$. Convex corners arise in graph theory, for instance as stable set polytopes of graphs. They are also natural objects of study in geometry, as they correspond to 1-unconditional norms in an obvious way. In this paper, we study a parameter of convex corners, which we call the {\em content}, that is related to the volume. This parameter has appeared implicitly before: both in geometry, chiefly in a paper of Meyer ({\em Israel J.\ Math.} 55 (1986) 317--327) effectively using content to give a proof of Saint-Raymond's Inequality on the volume product of a convex corner, and in combinatorics, especially in a paper of Sidorenko ({\em Order} 8 (1991) 331--340) relating content to the number of linear extensions of a partial order. One of our main aims is to expose connections between work in these two areas. We prove many new results, giving in particular various generalizations of Saint-Raymond's Inequality. Content also behaves well under the operation of {\em pointwise product} of two convex corners; our results enable us to give counter-examples to two conjectures of Bollob\'as and Leader ({\em Oper.\ Theory Adv.\ Appl.} 77 (1995) 13--24) on pointwise products. 1991 Mathematics Subject Classification: 52C07, 51M25, 52B11, 05C60, 06A07.

In this paper we consider Fourier multipliers for $L^p$ $(p>1)$ on Chébli-Trimèche hypergroups and establish a version of Hörmander's multiplier theorem. As applications we give some results concerning the Riesz potentials and oscillating multipliers.

1991 Mathematics Subject Classification: 43A62, 43A15, 43A32.

We derive Sobolev--Poincaré inequalities that estimate the $L^q(d\mu)$ norm of a function on a metric ball when $\mu$ is an arbitrary Borel measure. The estimate is in terms of the $L^1(d\nu)$ norm on the ball of a vector field gradient of the function, where $d\nu/dx$ is a power of a fractional maximal function of $\mu$. We show that the estimates are sharp in several senses, and we derive isoperimetric inequalities as corollaries. 1991 Mathematics Subject Classification: 46E35, 42B25.

Denote by $\phi_{\beta,p}$ the normal state on${\cal B}( \ell ^2 ({\Bbb N}))$ with list of eigenvalues $\lambda_{\beta ,p,k} =p^{-\beta k} (1-p^{-\beta} )$, where $k \in {\Bbb N}$, $\beta \in (0,1]$and $p$ is a prime number. For each $\beta $ and each subset ${\cal S}$ of the set ${\cal P}$ of all prime numbers, denote by $M_{\beta ,{\cal S}}$ the ITPFI factor (infinite tensor product of factors of type I)$$ \bigotimes\limits_{ p\in {\cal S}} \big( {\cal B}(\ell ^2 ({\Bbb N} )) ,\phi_{\beta,p} \big).$$ We prove that for any $\beta \in (0,1]$ and $\lambda \in [ 0,1]$, there exists ${\cal S}={\cal S}_{\beta,\lambda}$, asubset of ${\cal P}$, such that $M_{\beta, {\cal S}}$ isof type III$_\lambda$. Moreover, $M_{\beta ,{\cal P}}$ is the Araki--Woods factor $R_\infty$ for any $\beta \in (0,1]$. For any $\beta \in (0.5,1]$ and ${\cal S} \subset {\cal P}$, we show that $M_{\beta ,{\cal S}}$ is an ITPFI$_2$ factor. We study the class of Connes' T-groups of type III factors of the form $M_{\beta,{\cal S}}$, where ${\cal S} \subset {\cal P}$, which we call $\beta$-representable subgroups of ${\Bbb R}$ and show that it is rich, proving for each $\beta \in (0,1]$ that if $H$ is a countable subgroup of ${\Bbb R}$and $\Sigma$ a countable subset of ${\Bbb R} \setminus H$, there exists a $\beta$-representable group $\Gamma$ which contains $H$ and does not intersect $\Sigma$.Moreover, we prove that any $\beta_0$-representable group is $\beta$-representable if $1\geq \beta_0 \geq \max (\beta ,0.07)$ and that the class of $1$-representable groups is closed under the natural action of ${\Bbb R}^*_+$on the subgroups of ${\Bbb R}$. Our results show in particular that all hyperfinite type III$_\lambda$ factors, with $\lambda \in (0,1]$, and a large class of type III$_0$ ITPFI factors are isomorphic to group von Neumann algebras associated with some restricted adelic productsof affine motion groups of $p$-adic lines.

1991 Mathematics Subject Classification: 46L35, 11N05.