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.

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 $G$ 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.

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.

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.

We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (in other words, the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize perturbations which lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, or just quasi-equivalent, with Fock, coherent or quasifree representations. 1991 Mathematics Subject Classification: 81S05, 46L60, 81T05.

In recent years the theory of structured ring spectra (formerly known as A$_{\infty}$- and E$_{\infty}$-ring spectra) has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra. In this paper we provide a general method for constructing model category structures for categories of ring, algebra, and module spectra. This provides the necessary input for obtaining model categories of symmetric ring spectra, functors with smash product, $\Gamma$-rings, and diagram ring spectra. Algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra. 1991 Mathematics Subject Classification: primary 55U35; secondary 18D10.