Given that available technology permits one to establish that almost all natural numbers satisfying appropriate congruence conditions are represented as the sum of three squares of prime numbers, one expects strong estimates to be attainable for exceptional sets in the analogous problem involving sums of four squares of primes. Let E(N)$ denote the number of positive integers not exceeding N that are congruent to 4 modulo 24, yet cannot be written as the sum of four squares of prime numbers. A method is described that shows that for each positive number $\epsilon$, one has $E(N) \ll N^{13/30 + \epsilon}$, thereby exploiting effectively the 'excess' fourth square of a prime so as to improve the recent bound $E(N) \ll N^{13/15 + \epsilon}$ due to J. Liu and M.-C. Liu. It transpires that the ideas underlying this progress permit estimates for exceptional sets in a variety of additive problems to be significantly slimmed whenever sufficiently many excess variables are available. Such ideas are illustrated for several additional problems involving sums of four squares.

2000 Mathematical Subject Classification: 11P32, 11P05, 11P55.

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 prove that the Novikov assembly map for a group $\Gamma$ factorizes, in ‘low homological degree’, through the algebraic K-theory of its integral group ring. In homological degree 2, this answers a question posed by N. Higson and P. Julg. As a direct application, we prove that if $\Gamma$ is torsion-free and satisfies the Baum-Connes conjecture, then the homology group $H_{1}(\Gamma;\,\mathbb{Z})$ injects in $K_{1}(C^{*}_{r}\Gamma)$ and in $K_{1}^{\rm alg}(A)$, for any ring $A$ such that $\mathbb{Z}\Gamma\subseteq A\subseteq C^{*}_{r}\Gamma$. If moreover $B\Gamma$ is of dimension less than or equal to 4, then we show that $H_{2}(\Gamma;\,\mathbb{Z})$ injects in $K_{0}(C^{*}_{r}\Gamma)$ and in $K_{2}^{\rm alg}(A)/\Delta_{2}$, where $A$ is as before, and $\Delta_{2}$ is generated by the Steinberg symbols $\{\gamma,\,\gamma\}$, for $\gamma\in\Gamma$.

2000 Mathematical Subject Classification: primary 19D55, 19Kxx, 58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.

Philip Hall proved that the group operation in a finitely generated nilpotent group can be described by polynomials in the coordinates with respect to a Mal'cev basis. In this paper we generalize this result to the class of polycyclic groups. Using the construction of a semi-simple splitting for a polycylic group as introduced by Dan Segal, we prove that the group operation is described by polynomial and exponential functions. Hall's result provided a powerful route to the understanding of the connection between the abstract nilpotent group and associated topological and algebraic groups. In the same manner, we apply our result for polycyclic groups to illuminate various constructions, due to Steve Donkin and Andy Magid, of topological and algebraic groups associated with a polycyclic group.

2000 Mathematical Subject Classification: 20F22, 20G15, 22E05.

We use Nielsen methods to study generating sets of subgroups of groups that act on simplicial trees and give several applications. In particular, we exhibit an explicit bound for the complexity of acylindrical splittings of a finitely generated group in terms of its rank. This is applied to JSJ-splittings of word-hyperbolic groups and 3-manifolds. As a last application we construct examples of amalgamated products that show that there exists no non-trivial rank formula for amalgamated products.

2000 Mathematical Subject Classification: 20E06, 20E08, 57M27.

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.

Let $L$ denote the sub-Laplacian on the Heisenberg group $\mathbb{H}_n$ and $T_r^\lambda := (1 - r L)^\lambda_+$ the corresponding Bochner-Riesz operator. Let $Q$ denote the homogeneous dimension and $D$ the Euclidean dimension of $\mathbb{H}_n$. We prove convergence a.e. of the Bochner-Riesz means $T_r^\lambda f$ as $r \rightarrow 0$ for $\lambda > 0$ and for all $f \in L^p(\mathbb{H}_n)$, provided that \frac{Q-1}{Q} \Big(\frac{1}{2} - \frac{\lambda}{D-1} \Big) < 1/p \le 1/2. Our proof is based on explicit formulas for the operators $\partial_{\omega^a}$ with $a \in \mathbb{C}$, defined on the dual of $\mathbb{H}_n$ by $\partial_{\omega^a} \widehat{f} := \widehat{\omega^a f}$, which may be of independent interest. Here $\omega$ is given by $\omega(z,u) := |z|^2 - 4iu$ for all $(z,u) \in \mathbb{H}_n$.

2000 Mathematical Subject Classification: 22E30, 43A80.

We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C*-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism. These results are employed to obtain classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin, and to reprove the classification of purely infinite simple nuclear C*-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to the classification of nuclear C*-algebras.

2000 Mathematical Subject Classification: primary 46L35; secondary 19K14, 19K35, 46L80.

We extend the notion of Ribaucour transformation from classical surface theory to the theory of holonomic submanifolds of pseudo-Riemannian space forms with arbitrary dimension and codimension, that is, submanifolds with flat normal bundle admitting a global system of principal coordinates. Bianchi gave a positive answer to the question of whether among the Ribaucour transforms of a surface with constant mean or Gaussian curvature there exist other surfaces with the same property. Our main achievement is to solve the same problem for the class of holonomic submanifolds with constant sectional curvature.

2000 Mathematical Subject Classification: 53B25, 58J72.

This is the fifth in a series of papers constructing explicit examples of special Lagrangian submanifolds in ${\mathbb C}^m$. A submanifold of ${\mathbb C}^m$ is ruled if it is fibred by a family of real straight lines in ${\mathbb C}^m$. This paper studies ruled special Lagrangian 3-folds in ${\mathbb C}^3$, giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko, and Bryant. Special Lagrangian cones in ${\mathbb C}^3$ are automatically ruled, and each ruled special Lagrangian 3-fold is asymptotic to a unique special Lagrangian cone. We study the family of ruled special Lagrangian 3-folds N asymptotic to a fixed special Lagrangian cone N0. We find that this depends on solving a linear equation, so that the family of such N has the structure of a vector space. We also show that the intersection $\Sigma$ of N0 with the unit sphere ${\mathcal S}^5$ in ${\mathbb C}^3$ is a Riemann surface, and construct a ruled special Lagrangian 3-fold N asymptotic to N0 for each holomorphic vector field w on $\Sigma$. As corollaries of this we write down two large families of explicit special Lagrangian 3-folds in ${\mathbb C}^3$ depending on a holomorphic function on $\mathbb C$, which include many new examples of singularities of special Lagrangian 3-folds. We also show that each special Lagrangian T2-cone N0 can be extended to a 2-parameter family of ruled special Lagrangian 3-folds asymptotic to N0, and diffeomorphic to $T^2\times{\mathbb R}$.

2000 Mathematical Subject Classification: 53C38, 53D12.