The purpose of this article is to determine the representation type for all of the Hecke algebras of classical type. To do this, we combine methods from our previous work, which is used to obtain information on their Gabriel quivers, and recent advances in the theory of finite-dimensional algebras. The principal computation is for Hecke algebras of type B with two parameters. Then, we show that the representation type of Hecke algebras is governed by their Poincaré polynomials.

In this paper we study the diophantine equation of the title, which was first introduced by Nagell and Ljunggren during the first half of the twentieth century. We describe a method which allows us, on the one hand when n is fixed, to obtain an upper bound for q, and on the other hand when n and q are fixed, to obtain upper bounds for x and y which are far sharper than those derived from the theory of linear forms in logarithms. We also show how these bounds can be used even when they seem too large for a straightforward enumeration of the remaining possible values of x. By combining all these techniques, we are able to solve the equation in many cases, including the case when n has a prime divisor less than 13, or the case when n has a prime divisor which is less than or equal to 23 and distinct from q.

2000 Mathematical Subject Classification: primary 11D41; secondary 11J86, 11Y50.

It is shown that the system of two additive equations a_1 x_1^k + \ldots + a_s x_s^k = b_1 x_1^k + \ldots + b_s x_s^k =0 where $k \ge 2$ and $a_j$, $b_j$ are any given integers, has non-trivial solutions in all $p$-adic fields provided only that $s > 8k^2$. The constant 8 can be reduced when $k$ is not a power of 2. It is expected, in accordance with a classical conjecture of Artin, that the bound $8k^2$ can be replaced by $2k^2$.

2000 Mathematical Subject Classification:11D72.

Strong operator modules over a von Neumann algebra $R$ are introduced and the so-called extended Haagerup tensor product over $R$ of strong modules is studied. In the case $R={\bf C}$ and the spaces are weak* closed this product agrees with the weak* Haagerup tensor product of Blecher and Smith. If $C$ is the center of $R$, the extended Haagerup tensor product \rcr\ is a Banach algebra containing the central tensor product $R\haagc R$ of Chatterjee and Smith and has very nice properties concerning slice maps and the relative commutants of subalgebras. It is shown that \rcr\ is a dual Banach space, and all weak* closed two-sided ideals of the algebra \rcr\ are determined.

1991 Mathematics Subject Classification: 46L05

The Chowla–Selberg formula is applied in approximating a given Epstein zeta function. Partial sums of the series derive from the Chowla–Selberg formula, and although these partial sums satisfy a functional equation, as does an Epstein zeta function, they do not possess an Euler product. What we call partial sums throughout this paper may be considered as special cases concerning a more general function satisfying a functional equation only. In this article we study the distribution of zeros of the function. We show that in any strip containing the critical line, all but finitely many zeros of the function are simple and on the critical line. For many Epstein zeta functions we show that all but finitely many non-trivial zeros of partial sums in the Chowla–Selberg formula are simple and on the critical line.

A general selection theorem for a Vietoris-like hyperspace topology is obtained and, as a result, several known selection theorems for both `topology'- and `metric'-generated hyperspace topologies are improved from a common point of view. Applications are presented in characterizing classes of metrizable spaces by continuous selections for their hyperspaces.

1991 Mathematics Subject Classification: primary 54B20, 54C65; secondary 54E45, 54D65, 54E40, 54C10, 54G12.

This paper establishes recognition theorems for the finite Lie-type groups defined in odd characteristic $p$. The hypotheses of these theorems are couched in terms of certain local data in the form of an amalgam which is called a weak $BN$-pair of rank 2. The groups to be recognized are completions of these amalgams which satisfy the condition of being of local characteristic $p$ (a condition satisfied by the finite Lie-type groups defined in characteristic $p$). The results presented will find application in one of the ongoing revisions of the finite simple group classification.

We study the (restricted) holonomy group ${\rm Hol}(\nabla^{\perp})$ of the normal connection $\nabla^{\perp}$ (shortened to normal holonomy group) of a Kähler submanifold of a complex space form. We prove that if the normal holonomy group acts irreducibly on the normal space then it is linear isomorphic to the holonomy group of an irreducible Hermitian symmetric space. In particular, it is a compact group and the complex structure $J$ belongs to its Lie algebra.

We prove that the normal holonomy group acts irreducibly if the submanifold is full (that is, it is not contained in a totally geodesic proper Kähler submanifold) and the second fundamental form at some point has no kernel. For example, a Kähler–Einstein submanifold of $\mathbb{C} P^n$ has this property.

We define a new invariant $\mu$ of a Kähler submanifold of a complex space form. For non-full submanifolds, the invariant $\mu$ measures the deviation of $J$ from belonging to the normal holonomy algebra. For a Kähler–Einstein submanifold, the invariant $\mu$ is a rational function of the Einstein constant. By using the invariant $\mu$, we prove that the normal holonomy group of a not necessarily full Kähler–Einstein submanifold of $\mathbb{C} P^n$ is compact, and we give a list of possible holonomy groups.

The approach is based on a definition of the holonomy algebra ${\rm hol}(P)$ of an arbitrary curvature tensor field $P$ on a vector bundle with a connection and on a De Rham type decomposition theorem for ${\rm hol}(P)$.

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 of linear 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 intersection form multiplied by $(-1)^m$ is positive definite 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 a generalization of $E_6$. The lattice ${\mathcal L}^2 (X)$ is isomorphic to the laminated lattice of rank $22$. This isomorphism explains 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.

A theorem of K. W. Roggenkamp and L. L. Scott shows that for a finite group G with a normal p-subgroup containing its own centralizer, any two group bases of the integral group ring $\mathbb{Z} G$ are conjugate in the units of $\mathbb{Z}_{p}G$. Though the theorem presents itself in the work of others and appears to be needed, there is no published account. There seems to be a flaw in the proof, because a ‘theorem’ appearing in the survey [K. W. Roggenkamp, ‘The isomorphism problem for integral group rings of finite groups’, Progress in Mathematics 95 (Birkhäuser, Basel, 1991), pp. 193--220], where the main ingredients of a proof are given, is false. In this paper, it is shown how to close this gap, at least if one is only interested in the conclusion mentioned above. Therefore, some consequences of the results of A. Weiss on permutation modules are stated. The basic steps of which any proof should consist are discussed in some detail. In doing so, a complete, yet short, proof of the theorem is given in the case that G has a normal Sylow p-subgroup.

2000 Mathematical Subject Classification: primary 16U60; secondary 20C05.

Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a $2$-bridge knot. There is an invariant $\rho = p/q \in \mathbb{Q}/2 \mathbb{Z}$, with $p$ odd, defined for the pair $(K, \gamma)$.

The invariant $\rho$ has interesting geometric properties. It is often straightforward to calculate; for example, for $K$ a torus knot and $\gamma$ an annulus-spanning arc, $\rho(K, \gamma) = 1$. Although $\rho$ is defined abstractly, it is naturally revealed when $K \cup \gamma$ is put in thin position. If $\rho \neq 1$ then there is a minimal-genus Seifert surface $F$ for $K$ such that the tunnel $\gamma$ can be slid and isotoped to lie on $F$. One consequence is that if $\rho(K, \gamma) \neq 1$ then $\mathrm{genus}(K) > 1$. This confirms a conjecture of Goda and Teragaito for pairs $(K, \gamma)$ with $\rho(K, \gamma) \neq 1$.

In this paper Hill's equation $y''+qy=Ey$, where $q$ is a complex-valued function with inverse square singularities, is studied. Results on the dependence of solutions to initial value problems on the parameter $E$ and the initial point $x_0$, on the structure of the conditional stability set, and on the asymptotic distribution of (semi-)periodic and Sturm-Liouville eigenvalues are obtained. It is proved that a certain subset of the set of Floquet solutions is a line bundle on a certain analytic curve in ${\Bbb C}^2$. We establish necessary and sufficient conditions for $q$ to be algebro-geometric, that is, to be a stationary solution of some equation in the Korteweg-de Vries (KdV) hierarchy. To do this a distinction between movable and immovable Dirichlet eigenvalues is employed. Finally, an example showing that the finite-band property does not imply that $q$ is algebro-geometric is given. This is in contrast to the case where $q$ is real and non-singular.

1991 Mathematics Subject Classification: 34L40, 14H60.

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.

For systems of elliptic equations of Fitzhugh–Nagumo type on bounded domains and with small diffusion in one equation, we construct solutions with multiple sharp peaks close to each other and close to, but not on, the boundary. This is a striking contrast to results for scalar equations.

For some symmetric domains, we also construct similar multipeak solutions except that here the peaks are not close to each other.

In this paper we describe a tensor space representation of the blob algebra over a ring allowing base change to every interesting (that is, non-semisimple) specialisation. We show that in quasihereditary specialisations this passes to a full tilting module. We consider the nature of the resultant Ringel dual algebra.

Over a fixed finite field ${\bf F}_p$, families ofpolynomial equations$f_i(x_1, \dots, x_{n_N}) = 0$ for $i = 1, \dots, k_N$,that are uniformly determined by a parameter $N$, areconsidered. The notion of a uniform family is defined interms of first-order logic.A notion of an abstract Euler characteristic is used togive sense to a statement that the system has a solutionfor infinite $N$, and a statement linking the solvabilityof a linear system for infinite $N$ with its solvabilityfor finite $N$ is proved.This characterisation is used to formulate a criterionyielding degree lower bounds for various ideal membership proof systems (for example,Nullstellensatz and the polynomial calculus).Further, several results about Euler structures(structures with an abstract Euler characteristic) are proved, and the case of fields, in particular,is investigated more closely. 1991 Mathematics Subject Classification: primary 03F20, 12L12, 15A06;secondary 03C99, 12E12, 68Q15, 13L05.

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.

Motivated by complex oriented equivariant cohomology theories, we give a natural algebraic definition of an $A$-equivariant formal group law for any abelian compact Lie group $A$. The complex orientedcohomology of the classifying space for line bundles gives an example.We also show how the choice of a complete flag gives rise to a basis and a means of calculation. This allows us to deduce thatthere is a universal ring $L_A$ for $A$-equivariant formal group laws and that it is generated by the Euler classes and the coefficients of the coproduct of the orientation. Westudy a number of topological cases in some detail. 1991 Mathematics Subject Classification: 14L05, 55N22, 55N91, 57R85.

Let $F$ be a non-Archimedean local field. In an earlier paper, we constructed certain types (in the sense of Bushnell and Kutzko) in $SL_n (F)$, in fact enough to describe the non-supercuspidal components of the Bernstein decomposition of this group. We now give an almost fully explicit presentation of the Hecke algebras of these types.

Let $\varphi:A\to B$ be a homomorphism of finite-dimensional algebras over an algebraically closed field and $\varphi^{(c)}:{\rm mod}_B^c\to{\rm mod}_A^c$ the induced morphism of the associated module schemes for any integer $c\geq 1$. We prove that if the induced functor ${\rm mod\,} B\to{\rm mod\,} A$ is hom-controlled then the restriction of $\varphi^{(c)}$ to any connected component of ${\rm mod}_B^c$ is a composition of a smooth morphism followed by an immersion. Some new results on the types of singularities in the orbit closures of module schemes are also proved.

2000 Mathematical Subject Classification: 14B05, 14L30, 16G10.