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 andsufficient 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 isBorel-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 relationreducible to both $D$ and to $E$ has to be essentially countable. 2000 Mathematics Subject Classification: 03E15.

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 spacewhich 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 proofshows 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.

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, weobtain 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 anupper 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 whichnine cubes are required. 1991 Mathematics Subject Classification: 11P05, 11L15, 11P55.

We prove a bounded decomposition for higher order Hankel forms and characterize the first order Hochschild cohomology groups of the disk algebrawith 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 involvingmultilinear maps on the $n$-fold product of the disk algebra and we obtain several higher order analogues of the factorization results due to Aleksandrov andPeller. 2000 Mathematics Subject Classification: 47B35, 46E15, 46E25.

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 oflinear 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 intersectionformmultiplied by $(-1)^m$ is positivedefinite 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 ageneralization of $E_6$.The lattice ${\mathcal L}^2 (X)$ is isomorphic to the laminated lattice of rank $22$.This isomorphismexplains 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.

We investigate asphericity of the relative group presentation $$\langle G,t\mid atbtctdtet=1\rangle$$ and prove it aspherical provided thesubgroup 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\zetas^{-1}\rangle. 2000 Mathematics Subject Classification: 20F05, 57M05.

In this paper a $T(1)$ theorem for Calder\'on--Zygmundoperators (CZOs) valid for non-doubling measures with atoms is proved. In the classical Calder\'on--Zigmund theory an essential assumption is the doubling property of the underlying measure $\mu$ in ${\mathbb R}^n$. Some recent results have shown that a big part of the classical theory can be extended to the case of non-doubling measures, with only some mild `growth' condition on the measure $\mu$.In the particular case of the Cauchy transform $Cf(x) = \int f(y)/(y-x) \, d\mu(y)$, the main result of the paper can be stated easily: the Cauchy transform is bounded in $L^2(\mu)$ if and only if it is bounded in $L^2(\mu)$ over characteristic functions of squares.The proof of the main result is based on the use of the Haar basis with random dyadic lattices, following an idea of Nazarov, Treil and Volberg (`Cauchy integraland Calder\'on--Zygmund operators on nonhomogeneous spaces', {\em Internat. Math. Res. Notices} (1997) 15, 703--726), where a $T(1)$ theorem fornon-doubling continuous measures was obtained.The case of a measure $\mu$ with atoms studied in the present paper is quite different from the case ofcontinuous measures. For example, if $\mu$ has atoms, the $L^2(\mu)$ boundedness of a CZO does not imply its $L^p(\mu)$ boundedness for some $p\neq2$. The differences arise because a measure $\mu$ with atoms does notsatisfy the essential `growth' condition that is involved in the arguments for proving the $L^2$ boundedness of CZOs for continuous measures. 2000Mathematics Subject Classification: 42B20.

We construct a homological approximation to thepartition complex, and identify it as the Titsbuilding. This gives a homological relationshipbetween the symmetric group and the affine group,leads to a geometric tie between symmetric powersof spheres and the Steinberg idempotent, andallows us to use the self-duality of theSteinberg module to study layers in the Goodwillietower of the identity functor. 2000 Mathematics Subject Classification:55N25, 55S15, 20B30, 55P25.