A precise description of a subset $\mathcal{A}$ of $\mathbb{Z} / n \mathbb{Z}$ satisfying

$$ | \mathcal{A} + \mathcal{A} | \leq 2.04 | \mathcal{A} | $$

is given. Basically, there exists a subgroup $\mathcal{H}$ of $\mathbb{Z} / n \mathbb{Z}$ such that $\mathcal{A}$ is included in an arithmetic progression of $\ell$ cosets modulo $\mathcal{H}$ and

$$(\ell - 1) | \mathcal{H} | \leq | \mathcal{A} + \mathcal{A} | - | \mathcal{A} |.$$

2000 Mathematical Subject Classification: 11B50, 11B83, 20E34.

In this paper we give a realization of crystal bases for quantum affine algebras using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks with various shapes that are built on a given ground-state wall and can be viewed as generalizations of Young diagrams. The rules for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. The crystal graph of a basic representation is characterized as the set of all reduced proper Young walls. The character of a basic representation can be computed easily by counting the number of colored blocks that have been added to the ground-state wall.

2000 Mathematical Subject Classification: 17B37, 17B65, 81R50, 82B23.

Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameterized family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the ‘rational Cherednik algebra’, and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups $G(m, p, N)$, the set of singular parameters in the parameterized family of these structures is described explicitly, using the theory of non-symmetric Jack polynomials.

2000 Mathematical Subject Classification: 20F55 (primary), 52C35, 05E05, 33C08 (secondary).

Broué's abelian defect conjecture suggests a deep link between the module categories of a block of a group algebra and its Brauer correspondent, viz. that they should be derived equivalent. We are able to verify Broué's conjecture for the Hall–Janko group, even its double cover $2.J_2$, as well as for $U_3(4)$ and ${\rm Sp}_4(4)$. In fact we verify Rickard's refinement to Broué's conjecture and show that the derived equivalence can be chosen to be a splendid equivalence for these examples.

2000 Mathematical Subject Classification: 20C20, 20C34.

Let $D$ be a bounded strongly convex domain and let $f$ be a holomorphic self-map of $D$. In this paper we introduce and study the dilatation $\alpha (f)$ of $f$ defined, if $f$ has no fixed points in $D$, as the usual boundary dilatation coefficient of $f$ at its Wolff point, or, if $f$ has some fixed points in $D$, as the ratio of shrinking of the Kobayashi balls around a fixed point of $f$. In particular, we show that the map $\alpha$, defined as $\alpha : f \mapsto \alpha (f) \in [0,1]$, is lower semicontinuous. Among other things, this allows us to study the limits of a family of holomorphic self-maps of $D$. In the case of an inner fixed point, the dilatation is an intrinsic measure of the order of contact of $f(D)$ to $\partial D$.

Finally, using complex geodesics, we define and study a directional dilatation, which is a measure of the shrinking of the domain along a given direction. Again, results of semicontinuity are given and applied to a family of holomorphic self-maps.

2000 Mathematical Subject Classification: primary 32H99; secondary 30F99, 32H15.

The conjecture that Radon measures in Euclidean space with positive finite density are rectifiable was a central problem in Geometric Measure Theory for fifty years. This conjecture was positively resolved by Preiss in 1986, using methods entirely dependent on the symmetry of the Euclidean unit ball. Since then, due to reasons of isometric immersion of metric spaces into $l_{\infty}$ and the uncommon nature of the sup norm even in finite dimensions, a popular model problem for generalising this result to non-Euclidean spaces has been the study of 2-uniform measures in $l^{3}_{\infty}$. The rectifiability or otherwise of these measures has been a well-known question.

In this paper the stronger result that locally 2-uniform measures in $l^{3}_{\infty}$ are rectifiable is proved. This is the first result that proves rectifiability, from an initial condition about densities, for general Radon measures of dimension greater than 1 outside Euclidean space.

2000 Mathematical Subject Classification: 28A75.

We classify smooth ${\rm SO}(n)$-equivariant principal bundles over $S^n$ in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant $(\Pi, G)$-bundles over manifolds with cohomogeneity 1.

2000 Mathematical Subject Classification: 55R91.