Two locally generic maps $f, g \colon M^n \to \mathbb{R}^{2n - 1}$ are regularly homotopic if they lie in the same path-component of the space of locally generic maps. Our main result is that if $n \neq 3$ and $M^n$ is a closed $n$-manifold then the regular homotopy class of every locally generic map $f \colon M^n \to \mathbb{R}^{2n - 1}$ is completely determined by the number of its singular points provided that $f$ is singular (that is, $f$ is not an immersion).

A cover of a hypergraph is a collection of edges whose union contains all vertices. Let $H = (V, E)$ be a $k$-uniform, $D$-regular hypergraph on $n$ vertices, in which no two vertices are contained in more than $o(D / e^{2k} \log D)$ edges as $D$ tends to infinity. Our results include the fact that if $k = o(\log D)$, then there is a cover of $(1 + o(1)) n / k$ edges, extending the known result that this holds for fixed $k$. On the other hand, if $k \geq 4 \log D $ then there are $k$-uniform, $D$-regular hypergraphs on $n$ vertices in which no two vertices are contained in more than one edge, and yet the smallest cover has at least $\Omega (\frac {n}{k} \log (\frac {k}{\log D} ))$ edges. Several extensions and variants are also obtained, as well as the following geometric application. The minimum number of lines required to separate $n$ random points in the unit square is, almost surely, $\Theta (n^{2/3} / (\log n)^{1/3}).$

2000 Mathematical Subject Classification: 05C65, 05D15, 60D05.

A scheme of construction of infinite groups, other than simple groups, free groups of infinite rank and the infinite cyclic group, which are isomorphic to all their non-trivial normal subgroups is presented. Some results about the automorphism groups of simple infinite groups are also obtained. In particular, it is proved that there is an infinite group $G$ of any sufficiently large prime exponent $p$ (or which is torsion-free) all of whose proper subgroups are cyclic, and such that the groups ${\rm Aut}\,G$ and ${\rm Out}\,G$ are isomorphic. The proofs use the technique of graded diagrams developed by A. Yu. Ol'shanskii.

1991 Mathematics Subject Classification: 20F05, 20F06.

We present a vector field method for showing that certain subgroups of the mapping class group $\Gamma$ of a Riemann surface of infinite topological type act properly discontinuously. We apply the method to the group of homotopy classes of quasiconformal self-maps of the complement $\Omega$ of a Cantor set in $\mathbb{C}$. When the Cantor set has bounded geometric type, we show that $\Gamma(\Omega)$ acts on the Teichmüller space $T(\Omega)$ properly discontinuously. Also, we apply the same method to show that the pure mapping class group $\Gamma_0(\Omega \cup \{\infty\})$ acts properly discontinuously on $T(\Omega \cup \{\infty\})$.

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.

A finite Borel measure $\mu$ on $\mathbb R^d$ is called R-O varying with index $F$ if there exist a $\operatorname{GL}(\mathbb R^d)$-valued function $f$ varying regularly with index $(-F)$, an increasing function $k : (0,\infty)\to (0,\infty)$ with $k(t)\to\infty$ and $k(t+1)/k(t)\to c\geq 1$ as $t\to\infty$, and a $\sigma$-finite measure $\phi$ on $\mathbb R^d\setminus\{0\}$ such that \[k(t)\cdot(f(k(t))\mu)\to\phi\quad \text{as $t\to\infty$.}\] R-O varying measures generalize regularly varying measures introduced by Meerschaert (see M.~M.~Meerschaert, `Regular variation in $\mathbb R^k$', {\it Proc.\ Amer.\ Math.\ Soc.} 102 (1988) 341--348) and have numerous applications in limit theorems for probability measures. For an R-O varying measure $\mu$ and $-\infty<\infty$ let \begin{equation*} \begin{split} V_a(t,\theta) &= \int_{|\langle x,\theta\rangle|>t} |\langle x,\theta\rangle|^a\,d\mu(x),\\ U_b(t,\theta) &= \int_{|\langle x,\theta\rangle|\leq t} |\langle x,\theta\rangle|^b\,d\mu(x) \end{split} \end{equation*} denote the tail- and truncated moment functions of $\mu$ in the direction $\|\theta\|=1$. The purpose of this paper is to show that R-O variation of a measure implies sharp bounds on the growth rate of the tail- and truncated moment functions depending on the real parts of the eigenvalues of the index $F$ along a compact set of directions. Furthermore, bounds on the ratio of these functions for certain values of $a$ and $b$ are obtained. 1991 Mathematics Subject Classification: 60B10, 28C15.

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 set up a singularity-theoretic framework for classifying one-parameter steady-state bifurcations with hidden symmetries. This framework also permits a non-trivial linearization at the bifurcation point. Many problems can be reduced to this situation; for instance, the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with Neumann or Dirichlet boundary conditions. We formulate an appropriate equivalence relation with its associated tangent spaces, so that the usual methods of singularity theory become applicable. We also present an alternative method for computing those matrix-valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems. This result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an $N$-dimensional rectangle under Neumann boundary conditions.

1991 Mathematics Subject Classification: 58C27, 58F14.

This paper is devoted to the well-posedness of abstract Cauchy problems for quasi-linear evolution equations. The notion of Hadamard well-posedness is considered, and a new type of stability condition is introduced from the viewpoint of the theory of finite difference approximations. The result obtained here generalizes not only some results on abstract Cauchy problems closely related with the theory of integrated semigroups or regularized semigroups but also the Kato theorem on quasi-linear evolution equations. An application to some quasi-linear partial differential equation of weakly hyperbolic type is also given.

A d.c.e. (2-computably enumerable) degree d is pseudo-isolated if d itself is non-isolated (in the sense that no computably enumerable (c.e.) degree below d can bound the c.e. degrees below d) and there is a d.c.e. degree \textbf{b} < \textbf{d} bounding all c.e. degrees below d. We prove in this paper that the pseudo-isolated degrees are densely distributed in the c.e. degrees.

Over arbitrary fields of characteristic not equal to 2, we construct three families of simple Lie algebras and six families of simple Lie superalgebras of matrices with entries chosen from different one-sided ideals of a simple associative algebra. These families correspond to the classical Lie algebras and superalgebras. Our constructions intermix the structure of the associative algebra and the structure of the matrix algebra in an essential, compatible way. Many examples of simple associative algebras without an identity element arise as a by-product. The study of conformal algebras and superalgebras often involves matrix algebras over associative algebras such as Weyl algebras, and for that reason, we illustrate our constructions by taking various one-sided ideals from a Weyl algebra or a quantum torus.

Let $F$ be an algebraically closed field of characteristic $p$, and let $H$ be a finite group. A natural and important problem in representation theory is to classify the pairs $(G, D)$, where $G$ is a subgroup of $H$, and $D$ is an irreducible $FH$-module that remains irreducible when restricted to $G$. For example, if $H$ is an almost simple group or a central extension of an almost simple group, then one encounters this problem in the attempt to determine the maximal subgroups of the finite classical groups. We solve the above problem for Schur's double covers $H = {\widehat S}_n$ and ${\widehat A}_n$ of the symmetric and alternating groups, and $G$ a Young-type subgroup. The answer is given in terms of the combinatorics of restricted, $p$-strict partitions; such partitions are used to parameterize the irreducible spin representations of ${\widehat S}_n$ and ${\widehat A}_n$. We exploit the connections, recently developed by Ariki, Grojnowski, Brundan, Kleshchev, and others, between modular branching rules, crystal graphs of affine Kac–Moody algebras, and affine Hecke algebras and related objects.

We give an example of an affinoid curve without analytic continuation. We use this to produce an example of an affinoid morphism that cannot be flattened by a finite sequence of local blow-ups. Thus the global rigid analogue of Hironaka's complex analytic flattening theorem given by T. Gardener and H. Schoutens, in Theorem 2.3 of ‘Flattening and subanalytic sets in rigid analytic geometry’, Proc. London Math. Soc (3) 83 (2001) 681–707, is not true. Since this is a key step in the proof of the affinoid elimination theorem (loc. cit. Theorem 3.12), that proof contains a serious gap. We also give an example of an affinoid subset of the plane that is not the image under a proper rigid analytic map of a set that is globally semianalytic in the domain of that map. This clarifies the relationship among several natural categories of rigid subanalytic sets.

Certain classes of smoothly approximable structures — the class of affine covers of Lie geometries — are shown to have the amalgamation property. In particular, this shows that any affine cover of a Lie geometry has the small index property.

2000 Mathematical Subject Classification: 03C45.

Suppose $r=p^b$, where $p$ is a prime. Let $V$ be an $n$-dimensional ${\rm GF}(r)$-space and $G$ a subgroup of ${\rm A\Gamma L}(V) \cong {\rm A\Gamma L}(n,r)$ containing all translations and acting primitively on the set of vectors in $V$. Denote by $G_0$ the stabilizer in $G$ of the zero vector, so that $G_0 \le {\rm \Gamma L}(V) \cong {\rm \Gamma L}(n,r)$ and $G$ is the semidirect product of $V$ and $G_0$. Suppose that the generalized Fitting subgroup $F^*(G_0)$ of $G_0$ is an exceptional (twisted or untwisted, quasisimple) Chevalley group and that $\Gamma$ is a graph structure on $V$ on which $G$ acts primitively and distance transitively. The content of this paper is that then $G$ and $\Gamma$ are known. This result solves an open case in the outstanding problem of classifying all finite primitive distance-transitive groups. 2000 Mathematics Subject Classification: primary 20B25; secondary 05C25, 20Gxx, 05E30.

We make a detailed study of the relation of a euclidean convex region $\Omega \subset \mathbb C$ to $\mathrm{Dome} (\Omega)$. The dome is the relative boundary, in the upper halfspace model of hyperbolic space, of the hyperbolic convex hull of the complement of $\Omega$. The first result is to prove that the nearest point retraction $r: \Omega \to \mathrm{Dome} (\Omega)$ is 2-quasiconformal. The second is to establish precise estimates of the distortion of $r$ near $\partial \Omega$.

We determine explicitly all the stable manifolds for the stochastic flows on certain homogeneous spaces of a semisimple Lie group of non-compact type induced by Lévy processes on the Lie group.

2000 Mathematical Subject Classification: 58J65.

Let $K/k$ be a finite unramified Galois extension of number fields with Galois group $G$. This determines two homomorphisms from the ideal class group $\mathrm{Cl}_k$ of $k$: the capitulation map $\mathrm{Cl}_k \to \mathrm{Cl}_K$ and the Artin map $\mathrm{Cl}_k \twoheadrightarrow G^{\mathrm{ab}}$ onto the abelianization $G^{\mathrm{ab}}$ of $G$. We call (ker (capitulation), $\mathrm{Cl}_k$, Artin) the capitulation triple of $K / k$.

Artin's transition to group theory shows that any triple $(X, Y, \zeta)$ which arises in this way satisfies the group-theoretic property of being a transfer triple for $G$, defined as follows: there exist a group extension $A \rightarrowtail H \twoheadrightarrow G$ with $A$ finite abelian and an isomorphism $\eta : Y \stackrel{\sim}{\to} H^{\mathrm{ab}}$ such that $\eta (X)$ is the kernel of the transfer homomorphism $H^{\mathrm{ab}}\to A$, and $\zeta$ is the composite of $\eta$ with $H^{\mathrm{ab}}\to G^{\mathrm{ab}}$.

When $G$ is abelian, we show that a triple $(X, Y, \zeta)$ is a transfer triple for $G$ if and only if $|G| X = 0$ and $|G|$ divides $|X|$. Whether all transfer triples for $G$ can be realized arithmetically remains an unsolved problem.

Let $\cal V$ be a complete connected hyperbolic 3-manifold of finite volume, with Liouville measure $m$, geodesic flow $\Gamma_t$ and Brownian motion $Z_t$. Let $\omega$ be a smooth 1-form, closed in the cusps of $\cal V$. Then the limit laws as $t \to \infty$ of $(t\log t)^{-1/2}\int_0^t\omega(\Gamma_s)$ under $m$ and of $(t\log t)^{-1/2}\int_0^t\omega(Z_s)$ are calculated, and seen to be Gaussian, and equal. The geodesic flow case is studied via the Brownian case.

1991 Mathematics Subject Classification: 60J65, 58F17, 51M10.