Let G be a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p[ges ]0. In this paper, we classify all pairs (X, Y) of reductive subgroups of G which have a dense (X, Y)-double coset in G. In fact, we show that there is a dense (X, Y)-double coset in G precisely when G=XY is a factorisation. The possible factorisations that can occur have recently been determined in [11, Theorem A].

We now state our main result.

Let

(formula here)

be the germ of a finite (that is, proper with finite fibres) complex analytic morphism from a complex analytic normal surface onto an open neighbourhood U of the origin 0 in the complex plane C2. Let u and v be coordinates of C2 defined on U. We shall call the triple (π, u, v) the initial data.

Let Δ stand for the discriminant locus of the germ π, that is, the image by π of the critical locus Γ of π.

Let (Δα)α∈A be the branches of the discriminant locus Δ at O which are not the coordinate axes.

For each α ∈ A, we define a rational number dα by

(formula here)

where I(–, –) denotes the intersection number at 0 of complex analytic curves in C2. The set of rational numbers dα, for α ∈ A, is a finite subset D of the set of rational numbers Q. We shall call D the set of discriminantal ratios of the initial data (π, u, v). The interesting situation is when one of the two coordinates (u, v) is tangent to some branch of Δ, otherwise D = {1}. The definition of D depends not only on the choice of the two coordinates, but also on their ordering.

In this paper we prove that the set D is a topological invariant of the initial data (π, u, v) (in a sense that we shall define below) and we give several ways to compute it. These results are first steps in the understanding of the geometry of the discriminant locus. We shall also see the relation with the geometry of the critical locus.

Recently J. Coates and R. Greenberg introduced a new important class of extensions of local number fields with finite residue field which they call deeply ramified fields. These extensions play an essential role in their study of the arithmetics of abelian varieties over local fields with finite residue fields [1].

The first aim of this paper is to provide an ‘elementary’ treatment of deeply ramified extensions of local fields with arbitrary perfect residue fields using a method different from the original approach of Coates and Greenberg. Equivalent properties (1), (3)–(8) of deeply ramified extensions in Section 1 are due to them, and for their proofs a presentation of the different as an integral was involved. We translate the most important constructions into the language of the Hasse–Herbrand function (Section 1) and then apply the methods of [3, Chapter 3], where the Hasse–Herbrand function is defined in terms of the norm map. Some of the properties of deeply ramified extensions (two implications in the language of this paper) have been already studied by M. Matignon [7] and J. Fresnel, M. Matignon [5] for different purposes (see Remark 1.6 and the beginning of Section 2).

The second aim of this text is to expose relations among classes of deeply ramified extensions, that of arithmetically profinite extensions (Subsections 2.1–2.4) and that of p-adic Lie extensions (Subsections 2.5–2.7). For local fields with finite residue field we give in Subsection 2.2 an example of a Galois deeply ramified extension with infinite residue extension in which every Galois deeply ramified subextension is not arithmetically profinite; and in Subsection 2.4 we give an example of a Galois deeply ramified extension with finite residue field extension and a nondiscrete set of breaks (that means that this extension is not arithmetically profinite). The main result is that for local fields with finite residue field the class of Galois deeply ramified extensions with finite residue extension and a discrete set of breaks coincides with the class of infinite Galois arithmetically profinite extensions (Proposition 2.3). However, in the case of the fields with infinite residue fields Proposition 2.3 does not hold (Subsection 2.1). In Subsection 2.5 we construct an example that shows that the class of infinite Galois totally ramified arithmetically profinite extensions is strictly larger than the class of the most natural arithmetic origin – the class of totally ramified p-adic Lie extensions. Subsection 2.6 contains an example of a Galois deeply and totally ramified extension L of a local field F with finite residue field such that the norm group of L/E is of finite index in E*.

It is proved that if an algebra R over a field can be endowed with a pointed and finite-dimensional ℕn-filtration such that the associated ℕn-graded algebra T is semi-commutative, then R is left and right finitely partitive. In order to do this, a multi-variable Poincaré series for every finitely generated graded T-module is considered and it is shown that this Poincaré series is a rational function. The methods apply to some iterated Ore extensions such as quantum matrices and quantum Weyl algebras as well as to the quantized enveloping algebra of [sfr ][lfr ](ν+1).

It is known that the group of homeomorphisms of a non-compact, connected, metrizable manifold acts transitively on certain spaces of measures defined, among other properties, by their behaviour at the manifold's set of ‘ends’. As a generalization of previous work of Fathi in the realm of compact manifolds, it is shown that the given action admits a continuous section when restricted to ‘biregular’ homeomorphisms and measures. This result exhibits the isotropy groups of measure-preserving homeomorphisms as deformation retracts of the corresponding acting groups.

Thompson's famous theorems on singular values–diagonal elements of the orbit of an n×n matrix A under the action (1) U(n) [otimes ] U(n) where A is complex, (2) SO(n) [otimes ] SO(n), where A is real, (3) O(n) [otimes ] O(n) where A is real are fully examined. Coupled with Kostant's result, the real semi-simple Lie algebra [sfr ][ofr ]n, n yields (2) and hence (3) and the sufficient part (the hard part) of (1). In other words, the curious subtracted term(s) are well explained. Although the diagonal elements corresponding to (1) do not form a convex set in [Copf ]n, the projection of the diagonal elements into ℝn (or iℝn) is convex and the characterization of the projection is related to weak majorization. An elementary proof is given for this hidden convexity result. Equivalent statements in terms of the Hadamard product are also given. The real simple Lie algebra [sfr ][ufr ]n, n shows that such a convexity result fits into the framework of Kostant's result. Convexity properties and torus relations are studied. Thompson's results on the convex hull of matrices (complex or real) with prescribed singular values, as well as Hermitian matrices (real symmetric matrices) with prescribed eigenvalues, are generalized in the context of Lie theory. Also considered are the real simple Lie algebras [sfr ][ofr ]p, q and [sfr ][ofr ]p, q, p < q, which yield the rectangular cases. It is proved that the real part and the imaginary part of the diagonal elements of complex symmetric matrices with prescribed singular values are identical to a convex set in ℝn and the characterization is related to weak majorization. The convex hull of complex symmetric matrices and the convex hull of complex skew symmetric matrices with prescribed singular values are given. Some questions are asked.

The paper proves canonical isomorphisms between Spin Verlinde spaces, that is, spaces of global sections of a determinant line bundle over the moduli space of semistable Spinn-bundles over a smooth projective curve C, and the dual spaces of theta functions over Prym varieties of unramified double covers of C.

For a wide class of one-parameter families of Thue equations of arbitrary degree n[ges ]3 all solutions are determined if the parameter is sufficiently large. The result is based on the Lang–Waldschmidt conjecture, on the primitivity of the associated number fields and on an index bound, which does not depend on the coefficients. By applying the theory of Hilbertian fields and results on thin sets, primitivity is proved for almost all choices (in the sense of density) of the parameters.

Let {Xi}i[ges ]1 be a sequence of independent random variables taking the values ±1 with the probability ½, and let us set Sn = X1 + X2 +…+ Xn. A classical theorem of Hardy and Littlewood (1914) says that, for any C > 0 and for all n large enough, we have

formula here

with probability 1. In 1924, Khinchin showed that (1) can be replaced by a sharper inequality

formula here

for any ε > 0. In view of Khinchin's result, inequality (1) has long been considered as one of a rather historical value. However, the recent results on Brownian motion on Riemannian manifolds give a new insight into it. In this paper, we show that an analogue of (1), for the Brownian motion on Riemannian manifolds of the polynomial volume growth, is sharp and, therefore, cannot be replaced by an analogue of (2).

Let f:M→N be a stable map between orientable 4-manifolds, where M is closed and N is stably parallelisable. It is shown that the signature of M vanishes if and only if there exists a stable map g:M→N homotopic to f which has only fold and cusp singularities. This together with results of Ando and Èliašberg shows that, in this situation, the Thom polynomials are the only obstructions to the elimination of the singularities except for the fold singularity. Also studied are some topological properties (including those of the discriminant set) of stable maps between 4-manifolds with only Ak-type singularities.

A weight system is defined from the (multivariable) Conway potential function. We also show that it can be calculated recursively by using five axioms.

A Coxeter group $W$ is said to be rigid if, given any two Coxeter systems $(W, S)$ and $(W, S^\prime)$ , there is an automorphism $\rho: W \longrightarrow W$ such that $\rho(S) = S^\prime$ . The class of Coxeter systems $(W, S)$ for which the Coxeter graph $\Gamma_S$ is complete and has only odd edge labels is considered. (Such a system is said to be of type $K_n$ .) It is shown that if $W$ has a type $K_n$ system, then any other system for $W$ is also type $K_n$ . Moreover, the multiset of edge labels on $\Gamma_S$ and $\Gamma_{S^\prime}$ agree. In particular, if all but one of the edge labels of $\Gamma_S$ are identical, then $W$ is rigid.

We prove a lifting theorem (from any characteristic $p$ to characteristic zero) for holomorphic Picard modular forms of any weight and level. The proof makes use of the knowledge of the geometry of the toroidal compactification of the Picard modular surface.

First, the strong Stone–Weierstrass conjecture for postliminal ${\rm JB}^*$ -triples is proved. This is combined with the recent classification of ${\rm JB}^*$ -triples $A$ for which $\partial_e(A^*_1)$ is ${\rm weak}^*$ dense in $A^*_1$ , and the weak Stone–Weierstrass theorem for ${\rm JB}^*$ -triples is obtained. If $B$ is a ${\rm JB}^*$ -subtriple of a ${\rm JB}^*$ -triple $A$ such that $B$ separates $\overline{\partial_e(A^*_1)}\cup\{0\}$ , then $B=A$ .

Suppose that A is a pointed CW-complex. The paper looks at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits; this is determined by an ordinal number called the complexity of B. Studying the complexity leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CWA(X) of an arbitrary space X.

A new non-oscillatory theory is presented for higher order delay and neutral equations. The results include both singular and nonsingular problems.

This article introduces the notion of 2-ruled 4-folds: submanifolds of ${\mathbb{R}}^n$ fibred over a 2-fold $\Sigma$ by affine 2-planes. This is motivated by a paper by Joyce and previous work of the present author. A 2-ruled 4-fold $M$ is r-framed if an oriented basis is smoothly assigned to each fibre, and then we may write $M$ in terms of orthogonal smooth maps $\phi_1,\phi_2:\Sigma\rightarrow\mathcal{S}^{n-1}$ and a smooth map $\psi:\Sigma\rightarrow{\mathbb{R}}^n$. We focus on 2-ruled Cayley 4-folds in ${\mathbb{R}}^8$ as certain other calibrated 4-folds in ${\mathbb{R}}^7$ and ${\mathbb{R}}^8$ can be considered as special cases. The main result characterizes non-planar, r-framed, 2-ruled Cayley 4-folds, using a coupled system of nonlinear, first-order, partial differential equations that $\phi_1$ and $\phi_2$ satisfy, and another such equation on $\psi$ which is linear in $\psi$. We give a means of constructing 2-ruled Cayley 4-folds starting from particular 2-ruled Cayley cones using holomorphic vector fields. This is used to give explicit examples of ${\mathbin{\rm U}}(1)$-invariant 2-ruled Cayley 4-folds asymptotic to a ${\mathbin{\rm U}}(1)^3$-invariant 2-ruled Cayley cone. Examples are also given based on ruled calibrated 3-folds in ${\mathbb{C}}^3$ and ${\mathbb{R}}^7$ and complex cones in ${\mathbb{C}}^4$.

Two convex bodies K and K′ in Euclidean space [ ]n can be said to be in exceptional relative position if they have a common boundary point at which the linear hulls of their normal cones have a non-trivial intersection. It is proved that the set of rigid motions g for which K and gK′ are in exceptional relative position is of Haar measure zero. A similar result holds true if ‘exceptional relative position’ is defined via common supporting hyperplanes. Both results were conjectured by S. Glasauer; they have applications in integral geometry.

Let E(ℤ) = {einx}n∈ℤ denote the trigonometrical exponential system. It is well known that E(ℤ) forms an orthogonal basis in the space L2(0, 2π). In 1964, H. Landau discovered that the trigonometrical system has the following property: certain small perturbations of E(ℤ) yield exponential systems which are complete in L2 on any finite union of 2π-periodic translations of any interval (ε, 2π−ε), 0 < ε < π.

On the symbolic space endowed with a metric given by a Gibbs measure, it is shown that, for any invariant probability measure $\mu$ other than the given Gibbs measure, the set of $\mu$-generical points satisfies a ‘zero-infinity law’ (in particular, its Hausdorff and packing measure are infinite). This extends a result of R. Kaufman on Besicovitch–Eggleston sets, and applies to level sets of Birkhoff averages and certain subsets of self-similar sets.