Let $(X_n,d_n)$ be a sequence of finite metric spaces of uniformly bounded diameter. An equivalence relation $D$ on the product $\prod_n X_n$ defined by $\vec x\, D\,\vec y$ if and only if $\limsup_n d_n(x_n,y_n)=0$ is a {\em $c_0$-equality\/}. A systematic study is made of $c_0$-equalities and Borel reductions between them. Necessary and sufficient conditions, expressed in terms of combinatorial properties of metrics $d_n$, are obtained for a $c_0$-equality to be effectively reducible to the isomorphism relation of countable structures. It is proved that every Borel equivalence relation $E$ reducible to a $c_0$-equality $D$ either reduces a $c_0$-equality $D'$ additively reducible to $D$, or is Borel-reducible to the equality relation on countable sets of reals. An appropriately defined sequence of metrics provides a $c_0$-equality which is a turbulent orbit equivalence relation with no minimum turbulent equivalence relation reducible to it. This answers a question of Hjorth. It is also shown that, whenever $E$ is an $F_\sigma$-equivalence relation and $D$ is a $c_0$-equality, every Borel equivalence relation reducible to both $D$ and to $E$ has to be essentially countable. 2000 Mathematics Subject Classification: 03E15.

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.

A regular map of type $\{m,n\}$ is a 2-cell embedding of a graph in an orientable surface, with the property that for any two directed edges $e$ and $e'$ there exists an orientation-preserving automorphism of the embedding that takes $e$ onto $e'$, and in which the face length and the vertex valence are $m$ and $n$, respectively. Such maps are known to be in a one-to-one correspondence with torsion-free normal subgroups of the triangle groups $T(2,m,n)$. We first show that some of the known existence results about regular maps follow from residual finiteness of triangle groups. With the help of representations of triangle groups in special linear groups over algebraic extensions of ${\mathbb Z}$ we then constructively describe homomorphisms from $T(2,m,n)=\langle y,z|\ y^m=z^n=(yz)^2=1\rangle$ into finite groups of order at most $c^r$ where $c=c(m,n)$, such that no non-identity word of length at most $r$ in $x,y$ is mapped onto the identity. As an application, for any hyperbolic pair $\{m,n\}$ and any $r$ we construct a finite regular map of type $\{m,n\}$ of size at most $C^r$, such that every non-contractible closed curve on the supporting surface of the map intersects the embedded graph in more than $r$ points. We also show that this result is the best possible up to determining $C=C(m,n)$. For $r\ge m$ the graphs of the above regular maps are arc-transitive, of valence $n$, and of girth $m$; moreover, if each prime divisor of $m$ is larger than $2n$ then these graphs are non-Cayley. 2000 Mathematics Subject Classification: 05C10, 05C25, 20F99, 20H25.

In the context of Mackey functors we introduce a category which is analogous to the category of modules for a quasi-hereditary algebra which have a filtration by standard objects. Many of the constructions which work for quasi-hereditary algebras can be done in this new context. In particular, we construct an analogue of the `Ringel dual', which turns out here to be a standardly stratified algebra. The Mackey functors which play the role of the standard objects are constructed in the same way as functors which have been used previously in parametrizing the simple Mackey functors, but instead of using simple modules in their construction (as was done before) we use p-permutation modules. These Mackey functors are obtained as adjoints of the operations of forming the Brauer quotient and its dual. The filtrations which have these Mackey functors as their factors are closely related to the filtrations whose terms are the sum of induction maps from specified subgroups, or are the common kernel of restriction maps to these subgroups. These latter filtrations appear in Conlon's decomposition theorems for the Green ring, as well as in other places, where they arise quite naturally.

2000 Mathematics Subject Classification: primary 20C20; secondary 20J05, 19A22, 16G70, 16E60.

As is usual in prime number theory, write $$\psi(x,q,a)=\sum_{{n\le x}\atop{n\equiv a\pmod q}}\Lambda(n).$$ It is well known that when $q$ is close to $x$ the average value of $$V(x,q)=\sum_{{a=1}\atop{(a,q)=1}}^q \left|\psi(x,q,a)-{x\over{\phi(q)}}\right|^2$$ is about $x\log q$, and recently Friedlander and Goldston have shown that if $$U(x,q)=x\log q-x\left( \gamma+\log 2\pi+\sum_{p|q}\frac{\log p}{p-1} \right),$$ then the first moment of $V(x,q)-U(x,q)$ is small. In this memoir it is shown that the same is true for all moments. 2000 Mathematics Subject Classification: 11N13.

Suppose that $X$ is a Polish space which is not $\sigma$-compact. We prove that for every Borel colouring of $X^2$ by countably many colours, there exists a monochromatic rectangle with both sides closed and not $\sigma$-compact. Moreover, every Borel colouring of $[X]^2$ by finitely many colours has a homogeneous set which is closed and not $\sigma$-compact. We also show that every Borel measurable function $f:X^2 \rightarrow X$ has a free set which is closed and not $\sigma$-compact. As corollaries of the proofs we obtain two results: firstly, the product forcing of two copies of superperfect tree forcing does not add a Cohen real, and, secondly, it is consistent with ZFC to have a closed subset of the Baire space which is not $\sigma$-compact and has the property that, for any three of its elements, none of them is constructible from the other two. A similar proof shows that it is consistent to have a Laver tree such that none of its branches is constructible from any other branch. The last four results answer questions of Goldstern and Brendle. 2000 Mathematics Subject Classification: 03E15, 26B99, 54H05.

The largest prime factor of $X^{3}+2$ was investigated in 1978 by Hooley, who gave a conditional proof that it is infinitely often at least as large as $X^{1+\delta}$, with a certain positive constant $\delta$. It is trivial to obtain such a result with $\delta=0$. One may think of Hooley's result as an approximation to the conjecture that $X^{3}+2$ is infinitely often prime. The condition required by Hooley, his R$^{*}$ conjecture, gives a non-trivial bound for short Ramanujan--Kloosterman sums. The present paper gives an unconditional proof that the largest prime factor of $X^{3}+2$ is infinitely often at least as large as $X^{1+\delta}$, though with a much smaller constant than that obtained by Hooley. In order to do this we prove a non-trivial bound for short Ramanujan--Kloosterman sums with smooth modulus. It is also necessary to modify the Chebychev method, as used by Hooley, so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus. 2000 Mathematics Subject Classification: 11N32.

In this paper we define and study Hopf C$^*$-algebras. Roughly speaking, a Hopf C$^*$-algebra is a C$^*$-algebra $A$ with a comultiplication $\phi : A \rightarrow M(A \otimes A)$ such that the maps $a \otimes b \mapsto \phi(a)(1 \otimes b)$ and $a \otimes b \mapsto (a \otimes 1)\phi(b)$ have their range in $A \otimes A$ and are injective after being extended to a larger natural domain, the Haagerup tensor product $A \otimes_h A$. In a purely algebraic setting, these conditions on $\phi$ are closely related to the existence of a counit and antipode. In this topological context, things turn out to be much more subtle, but nevertheless one can show the existence of a suitable counit and antipode under these conditions.

The basic example is the C$^*$-algebra $C_0(G)$ of continuous complex functions tending to zero at infinity on a locally compact group where the comultiplication is obtained by dualizing the group multiplication. But also the reduced group C$^*$-algebra $C^*_r(G)$ of a locally compact group with the well-known comultiplication falls in this category. In fact all locally compact quantum groups in the sense of Kustermans and the first author (such as the compact and discrete ones) as well as most of the known examples are included.

This theory differs from other similar approaches in that there is no Haar measure assumed.

2000 Mathematics Subject Classification: 46L65, 46L07, 46L89.

In an earlier paper the second author used the formal, algebraic properties of $2$-dimensional Shintani generating functions to construct a $1$-cocycle on ${\rm PGL}_2(\mathbb{Q})$. We aim to generalise these results by using such functions in dimension $n$ to obtain an $(n-1)$-cocycle on ${\rm PGL}_n(\mathbb{Q})$, presumably related to the Bernoulli and Eisenstein cocycles of R.~Sczech. By improving our methods we achieve this goal for $n=3$. For $n>3$ we encounter obstacles related to degenerate configurations of hyperplanes in $n$-space. Nevertheless, we obtain partial results closely connected to reciprocity laws for certain $n$-dimensional Dedekind sums. 1991 Mathematics Subject Classification: 11F20, 11F75.

Non-trivial estimates for fractional moments of smooth cubic Weyl sums are developed. Complemented by bounds for such sums of use on both the major and minor arcs in a Hardy--Littlewood dissection, these estimates are applied to derive an upper bound for the $s$th moment of the smooth cubic Weyl sum of the expected order of magnitude as soon as $s\ge 7.691$. Related arguments demonstrate that all large integers $n$ are represented as the sum of eight cubes of natural numbers, all of whose prime divisors are at most $\exp (c(\log n\log \log n)^{1/2})$, for a suitable positive number $c$. This conclusion improves a previous result of G. Harcos in which nine cubes are required. 1991 Mathematics Subject Classification: 11P05, 11L15, 11P55.

In 1914 Bohr proved that there is an $r \in (0,1)$ such that if a power series converges in the unit disk and its sum has modulus less than $1$ then, for $|z| < r$, the sum of absolute values of its terms is again less than $1$. Recently, analogous results have been obtained for functions of several variables. The aim of this paper is to place the theorem of Bohr in the context of solutions to second-order elliptic equations satisfying the maximum principle.

2000 Mathematics Subject Classification: 35J15, 32A05, 46A35.

Given any sequence of non-abelian finite simple primitive permutation groups $(S_{n})$, we construct a finitely generated group $G$ whose profinite completion is the infinite permutational wreath product $\ldots S_{n}\wr S_{n-1}\wr\ldots\wr S_{0}$. It follows that the upper composition factors of $G$ are exactly the groups $S_{n}$. By suitably choosing the sequence $(S_{n})$ we can arrange that $G$ has any one of a continuous range of slow, non-polynomial subgroup growth types. We also construct a $61$-generator perfect group that has every non-abelian finite simple group as a quotient. 2000 Mathematics Subject Classification: 20E07, 20E08, 20E18, 20E32.

We prove a bounded decomposition for higher order Hankel forms and characterize the first order Hochschild cohomology groups of the disk algebra with coefficients in the space of bounded Hankel forms of some fixed order. Although these groups are non-trivial, we prove that every bounded derivation is inner and necessarily implemented by a Hankel form of order one higher. In terms of operators, this result extends the similarity result of Aleksandrov and Peller. Both of the main structural theorems here rely on estimates involving multilinear maps on the $n$-fold product of the disk algebra and we obtain several higher order analogues of the factorization results due to Aleksandrov and Peller. 2000 Mathematics Subject Classification: 47B35, 46E15, 46E25.

Let $G$ be an almost simple algebraic group defined over ${\Bbb F}_p$ for some prime $p$. Denote by $G_1$ the first Frobenius kernel in $G$ and let $T$ be a maximal torus. In this paper we study certain Jantzen type filtrations on various modules in the representation theory of $G_1T$. We have such filtrations on the baby Verma modules $Z(\lambda)$, where $\lambda$ is a character of $T$. They are obtained via a certain deformation of the natural homomorphism from $Z(\lambda)$ into its contravariant dual $Z(\lambda)^\tau$. Using the same deformation we construct for each projective $G_1T$-module $Q$ a filtration of the vector space $F_\lambda(Q)=\text{Hom}_{G_1T}(Z(\lambda)^\tau, Q)$. We then prove that this filtration may also be described in terms of the above-mentioned homomorphism $Z(\lambda) \rightarrow Z(\lambda)^\tau$ and this leads us to a sum formula for our filtrations. When $Q$ is indecomposable with highest weight in the bottom alcove (with respect to some special point) we are able to compute the filtrations on $F_\lambda(Q)$ explicitly for all $\lambda$. This is then the starting point of an induction which proceeds via wall crossings to higher alcoves. If our filtrations behave as expected under such wall crossings then we obtain a precise relation between the dimensions of the layers in the filtrations of $F_\lambda (Q)$ for an arbitrary indecomposable projective $Q$ and the coefficients in the corresponding Kazhdan--Lusztig polynomials. We conclude the paper by proving that the above results in the $G_1T$ theory have some analogues in the representation theory of $G$ (where, however, we have to work with representations of bounded highest weights) and the corresponding theory for quantum groups at roots of unity. These results extend previous work by the first author. 2000 Mathematics Subject Classification: 20G05, 20G10, 17B37.

A generalized quadrangle $\cal S$ is laxly embedded in a (finite) projective space {\bf PG}$(d,q)$ if $\cal S$ is a subgeometry of the geometry of points and lines of {\bf PG}$(d,q)$, with the only condition that the points of $\cal S$ generate the whole space {\bf PG}$(d,q)$ (which one can always assume without loss of generality). In this paper, we classify thick laxly embedded quadrangles satisfying some additional hypotheses. The hypotheses are (a combination of) a restriction on the dimension $d$, a restriction on the parameters of $\cal S$, and an assumption on the isomorphism class of $\cal S$. In particular, the classification is complete in the following cases: \begin{enumerate} \item[(1)] for $d\geq 5$; \item[(2)] for $d=4$ and $\cal S$ having `known' order $(s,t)$ with $t\not= s^2$; \item[(3)] for $d\geq 3$ and $\cal S$ isomorphic to a finite Moufang quadrangle distinct from $W(s)$ with $s$ odd. \end{enumerate}

As a by-product, we obtain a new characterization theorem of the classical quadrangle $H(4,s^2)$, and we also show that every generalized quadrangle of order $(s,s+2)$, with $s>2$, has at least one non-regular line.

2000 Mathematics Subject Classification: 51E12.

Working in the category $\mathcal{T}$ of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functors $\mathcal{D}\longrightarrow \mathcal{T}$ for a suitable small topological category $\mathcal{D}$. When $\mathcal{D}$ is symmetric monoidal, there is a smash product that gives the category of $\mathcal{D}$-spaces a symmetric monoidal structure. Examples include \begin{enumerate} \item[] prespectra, as defined classically, \item[] symmetric spectra, as defined by Jeff Smith, \item[] orthogonal spectra, a coordinate-free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category, \item[] $\Gamma$-spaces, as defined by Graeme Segal, \item[] $\mathcal{W}$-spaces, an analogue of $\Gamma$-spaces with finite sets replaced by finite CW complexes in the domain category. \end{enumerate} We construct and compare model structures on these categories. With the caveat that $\Gamma$-spaces are always connective, these categories, and their simplicial analogues, are Quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category. Monoids in these categories are (strict) ring spectra. Often the subcategories of ring spectra, module spectra over a ring spectrum, and commutative ring spectra are also model categories. When this holds, the respective categories of ring and module spectra are Quillen equivalent and thus have equivalent homotopy categories. This allows interchangeable use of these categories in applications.

2000 Mathematics Subject Classification: primary 55P42; secondary 18A25, 18E30, 55U35.

Let $G$ be a semisimple algebraic group defined over an algebraically closed field $K$ of good characteristic $p>0.$ Let $u$ be a unipotent element of $G$ of order $p^{t}$, for some $t\in {\Bbb N}$. In this paper it is shown that $u$ lies in a closed subgroup of $G$ isomorphic to the {\it Witt group} $W_{t}(K),$ which is a $t$-dimensional connected abelian unipotent algebraic group. 2000 Mathematics Subject Classification: 20G15.

Let $X$ be the Fermat hypersurface of dimension $2m$ and of degree $q+1$ defined over an algebraically closed field of characteristic $p>0$, where $q$ is a power of $p$, and let $NL^m (X)$ be the free abelian group of numerical equivalence classes of linear subspaces of dimension $m$ contained in $X$. By the intersection form, we regard $NL^m (X)$ as a lattice. Investigating the configuration of these linear subspaces, we show that the rank of $NL^m (X)$ is equal to the $2m$th Betti number of $X$, that the intersection form multiplied by $(-1)^m$ is positive definite on the primitive part of $NL^m (X)$, and that the discriminant of $NL^m (X)$ is a power of~$p$. Let ${\mathcal L}^m (X)$ be the primitive part of $NL^m (X)$ equipped with the intersection form multiplied by $(-1)^m$. In the case $p=q=2$, the lattice ${\mathcal L}^m (X)$ is described in terms of certain codes associated with the unitary geometry over ${\mathbb F}_2$. Since ${\mathcal L}^1 (X)$ is isomorphic to the root lattice of type $E_6$, the series of lattices ${\mathcal L}^m (X)$ can be considered as a generalization of $E_6$. The lattice ${\mathcal L}^2 (X)$ is isomorphic to the laminated lattice of rank $22$. This isomorphism explains Conway's identification $\cdot 222\cong {\rm PSU}(6,2)$ geometrically. The lattice ${\mathcal L}^3 (X)$ is of discriminant $2^{16}\cdot 3$, minimal norm $8$, and kissing number $109421928$. 2000 Mathematics Subject Classification: 14C25, 11H31, 51D25.

Let $G$ be a $\sigma$-compact, locally compact group and $\mathcal I$ be a closed 2-sided ideal with finite codimension in $L^1(G)$. It is shown that there are a closed left ideal ${\mathcal L}$ having a right bounded approximate identity and a closed right ideal ${\mathcal R}$ having a left bounded approximate identity such that ${\mathcal I} = {\mathcal L} + {\mathcal R}$. The proof uses ideas from the theory of boundaries of random walks on groups. 2000 Mathematics Subject Classification: primary 43A20; secondary 42A85, 43A07, 46H10, 46H40, 60B11.

We investigate asphericity of the relative group presentation $$\langle G,t\mid atbtctdtet=1\rangle$$ and prove it aspherical provided the subgroup of $G$ generated by $$\{ ab^{-1}, bc^{-1}, cd^{-1}, de^{-1}\}$$ is neither finite cyclic nor a finite triangle group. We also prove a similar result for the closely related relative group presentation $$\langle G,s,t \mid \alpha s\beta s\gamma t= 1=\delta t\varepsilon t\zeta s^{-1}\rangle. 2000 Mathematics Subject Classification: 20F05, 57M05.