We prove that every infinite-dimensional Banach space $X$ having a
(not necessarily equivalent) real-analytic norm is real-analytic
diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an
infinite-dimensional Banach space and $F$ is a closed subspace of $X$
such that there is a real-analytic seminorm on $X$ whose set of zeros
is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus
F$ are real-analytic diffeomorphic. As an application we show the
existence of real-analytic free actions of the circle and the
$n$-torus on certain Banach spaces.

In this paper we extend a well-known theorem of M.~Scheunert on
skew-symmetric bicharacters of groups to the case of skew-symmetric
bicharacters on arbitrary cocommutative Hopf algebras over a field of
characteristic not 2. We also classify polycharacters on (restricted)
enveloping algebras and bicharacters on divided power algebras.

If an operator $T$ satisfies a modular inequality on a rearrangement
invariant space $L^\rho (\Omega,\mu)$, and if $p$ is strictly between
the indices of the space, then the Lebesgue inequality $\int |Tf|^p
\leq C \int |f|^p$ holds. This extrapolation result is a partial
converse to the usual interpolation results. A modular inequality for
Orlicz spaces takes the form $\int \Phi (|Tf|) \leq \int \Phi (C
|f|)$, and here, one can extrapolate to the (finite) indices $i(\Phi)$
and $I(\Phi)$ as well.

Let $G$ be a discrete subgroup of $\SL(2,\R)$ which contains
$\Gamma(N)$ for some $N$. If the genus of $X(G)$ is zero, then there
is a unique normalised generator of the field of $G$-automorphic
functions which is known as a normalised Hauptmodul. This paper gives
a characterisation of normalised Hauptmoduls as formal $q$ series
using modular polynomials.

A local version of $\VMO$ is defined, and the local Hardy space
$\hone$ is shown to be its dual. An application to weak-$*$
convergence in $\hone$ is proved.

Gromov introduced the concept of uniform embedding into Hilbert space
and asked if every separable metric space admits a uniform embedding
into Hilbert space. In this paper, we study uniform embedding into
Hilbert space and answer Gromov's question negatively.

We show that the set of values of an additive polynomial in several
variables with arguments in a formal Laurent series field over a
finite field has the optimal approximation property: every element in
the field has a (not necessarily unique) closest approximation in this
set of values. The approximation is with respect to the canonical
valuation on the field. This property is elementary in the language
of valued rings.

Let $k$ be a cyclic extension of odd prime degree $p$ of the field of
rational numbers. If $t$ denotes the number of primes that ramify in $k$,
it is known that the Hilbert $p$-class field tower of $k$ is infinite if
$t>3+2\sqrt p$. For each $t>2+\sqrt p$, this paper shows that a positive
proportion of such fields $k$ have infinite Hilbert $p$-class field towers.

Using a classical generalization of Jacobi's derivative formula, we
give an explicit expression for Gunning's prime form in genus 2 in
terms of theta functions and their derivatives.

We study ergodic properties of a family of interval maps that are
given as the fractional parts of certain real M\"obius
transformations. Included are the maps that are exactly
$n$-to-$1$, the classical Gauss map and the Renyi or backward
continued fraction map. A new approach is presented for deriving
explicit realizations of natural automorphic extensions and their
invariant measures.

D.~H.~Lehmer initiated the study of the distribution of totatives, which
are numbers coprime with a given integer. This led to various problems
considered by P.~Erd\H os, who made a conjecture on such distributions.
We prove his conjecture by establishing a theorem on the ordering of
residues.

Let $b$ be an integer with $b>1$. In this note, we prove that the
number of non-zero digits in the base $b$ representation of $n!$
grows at least as fast as a constant, depending on $b$, times $\log
n$.

Sufficient conditions are given in order to prove the lowest unknown case
of the grade conjecture. The proof combines vanishing results of local
cohomology and the $S_{2}$ condition.

We give a new measure-theoretical proof of the uniform distribution
property of points in model sets (cut and project sets). Each model
set comes as a member of a family of related model sets, obtained by
joint translation in its ambient (the `physical') space and its
internal space. We prove, assuming only that the window defining the
model set is measurable with compact closure, that almost surely the
distribution of points in any model set from such a family is uniform
in the sense of Weyl, and almost surely the model set is pure point
diffractive.

We give a topological interpretation of the core group invariant of a
surface embedded in $S^4$ \cite{F-R}, \cite{Ro}. We show that the
group is isomorphic to the free product of the fundamental group of
the double branch cover of $S^4$ with the surface as a branched set,
and the infinite cyclic group. We present a generalization for
unoriented surfaces, for other cyclic branched covers, and other
codimension two embeddings of manifolds in spheres.

Let $X^5 + aX + b \in Z[X]$ have Galois group $D_5$. Let $\theta$ be
a root of $X^5 + aX + b$. An explicit formula is given for the
discriminant of $Q(\theta)$.

Applications of minimal surface methods are made to obtain information
about univalent harmonic mappings. In the case where the mapping arises
as the Poisson integral of a step function, lower bounds for the number
of zeros of the dilatation are obtained in terms of the geometry of the
image.

Si $M$ est une vari\'et\'e de dimension $n$, compacte non simplement
connexe, on caract\'erise les m\'etriques riemanniennes sur $M$ dont
la fonction croissance a exactement deux singularit\'es.

We explicitly describe, in terms of indecomposable $\mathbb{Z}_2
[G]$-modules, the Galois module structure of ideals in totally
ramified biquadratic extensions of local number fields with only
one break in their ramification filtration. This paper completes
work begun in [Elder: Canad. J.~Math. (5) {\bf 50}(1998), 1007--1047].

It is known that the $K$-theory of a large class of groups can be
computed from the $K$-theory of their virtually infinite cyclic
subgroups. On the other hand, Nil-groups appear to be the obstacle in
calculations involving the $K$-theory of the latter. The main
difficulty in the calculation of Nil-groups is that they are
infinitely generated when they do not vanish. We develop methods for
computing the exponent of ${\nk}_0$-groups that appear in the
calculation of the $K_0$-groups of virtually infinite cyclic groups.

We prove that the Mahler measure of an algebraic number cannot be too
close to an integer, unless we have equality. The examples of certain
Pisot numbers show that the respective inequality is sharp up to a
constant. All cases when the measure is equal to the integer are
described in terms of the minimal polynomials.

We compute the rational Chow groups of supersingular abelian varieties
and some other related varieties, such as supersingular Fermat
varieties and supersingular $K3$ surfaces. These computations are
concordant with the conjectural relationship, for a smooth projective
variety, between the structure of Chow groups and the coniveau
filtration on the cohomology.

The Griffiths group $\Gr^r(X)$ of a smooth projective variety $X$ over
an algebraically closed field is defined to be the group of homologically
trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of
algebraically trivial algebraic cycles. The main result of this paper is
that the Griffiths group $\Gr^2 (A_{\bar{k}})$ of a supersingular
abelian variety $A_{\bar{k}}$ over the algebraic closure of a finite
field of characteristic $p$ is at most a $p$-primary torsion group.
As a corollary the same conclusion holds for supersingular Fermat
threefolds. In contrast, using methods of C.~Schoen it is also
shown that if the Tate conjecture is valid for all smooth
projective surfaces and all finite extensions of the finite ground
field $k$ of characteristic $p>2$, then the Griffiths group of any ordinary
abelian threefold $A_{\bar{k}}$ over the algebraic closure of $k$ is
non-trivial; in fact, for all but a finite number of primes $\ell\ne p$ it
is the case that $\Gr^2 (A_{\bar{k}}) \otimes \Z_\ell \neq 0$.

An irreducible supercuspidal representation $\pi$ of $G=
\GL(n,F)$, where $F$ is a nonarchimedean local field of
characteristic zero, is said to be ``distinguished'' by a
subgroup $H$ of $G$ and a quasicharacter $\chi$ of $H$ if
$\Hom_H(\pi,\chi)\noteq 0$. There is a suitable global analogue
of this notion for and irreducible, automorphic, cuspidal
representation associated to $\GL(n)$. Under certain general
hypotheses, it is shown in this paper that every distinguished,
irreducible, supercuspidal representation may be realized as a
local component of a distinguished, irreducible automorphic,
cuspidal representation. Applications to the theory of
distinguished supercuspidal representations are provided.

The geometry of indicatrices is the foundation of Minkowski geometry.
A strongly convex indicatrix in a vector space is a strongly convex
hypersurface. It admits a Riemannian metric and has a distinguished
invariant---(Cartan) torsion. We prove the existence of non-trivial
strongly convex indicatrices with vanishing mean torsion and discuss
the relationship between the mean torsion and the Riemannian curvature
tensor for indicatrices of Randers type.

We combine the deep methods of Frey, Ribet, Serre and Wiles with some
results of Darmon, Merel and Poonen to solve certain explicit
diophantine equations. In particular, we prove that the area of a
primitive Pythagorean triangle is never a perfect power, and that each
of the equations $X^4 - 4Y^4 = Z^p$, $X^4 + 4Y^p = Z^2$ has no
non-trivial solution. Proofs are short and rest heavily on results
whose proofs required Wiles' deep machinery.

We use the theory of Jacobi-like forms to construct modular forms for a
congruence subgroup of $\SL(2,\mathbb{R})$ which can be expressed as linear
combinations of products of certain theta functions.

H.~O.~Kim has shown that contrary to the case of
$H^p$-space, the Smirnov class $M$ defined by the radial maximal
function is essentially smaller than the classical Smirnov class
of the disk. In the paper we show that these two classes have the
same corresponding locally convex structure, {\it i.e.} they have the
same dual spaces and the same Fr\'echet envelopes. We describe a
general form of a continuous linear functional on $M$ and
multiplier from $M$ into $H^p$, $0 < p \leq \infty$.

In this note we show that the image of the transfer for permutation
representations of finite groups is generated by the transfers of
special monomials. This leads to a description of the image of the
transfer of the alternating groups. We also determine the height of
these ideals.

A new construction of the ordinary residue of differential forms is
given. This construction is intrinsic, \ie, it is defined without
local coordinates, and it is geometric: it is constructed out of the
geometric structure of the local and global cohomology groups of the
differentials. The Residue Theorem and the local calculation then
follow from geometric reasons.

We study a class of subgroups of $\PSL_2 (\mathbb{Z})$ which can be
characterized in different ways, such as congruence groups, modular
forms, modular curves, elliptic surfaces, lattices and graphs.

A well-known theorem of Sarason [11] asserts that if $[T_f,T_h]$ is
compact for every $h \in H^\infty$, then $f \in H^\infty + C(T)$.
Using local analysis in the full Toeplitz algebra $\calT = \calT
(L^\infty)$, we show that the membership $f \in H^\infty + C(T)$ can
be inferred from the compactness of a much smaller collection of
commutators $[T_f,T_h]$. Using this strengthened result and a theorem
of Davidson [2], we construct a proper $C^\ast$-subalgebra $\calT
(\calL)$ of $\calT$ which has the same essential commutant as that of
$\calT$. Thus the image of $\calT (\calL)$ in the Calkin algebra does
not satisfy the double commutant relation [12], [1]. We will also
show that no {\it separable} subalgebra $\calS$ of $\calT$ is capable
of conferring the membership $f \in H^\infty + C(T)$ through the
compactness of the commutators $\{[T_f,S] : S \in \calS\}$.

The Cuntz-Krieger algebra $\mathcal{O}_B$ is defined for an
arbitrary, possibly infinite and infinite valued, matrix $B$. A graph
$C^{\ast}$-algebra $G^{\ast} (E)$ is introduced for an arbitrary
directed graph $E$, and is shown to coincide with a previously defined
graph algebra $C^{\ast} (E)$ if each source of $E$ emits only finitely
many edges. Each graph algebra $G^{\ast} (E)$ is isomorphic to the
Cuntz-Krieger algebra $\mathcal{O}_B$ where $B$ is the vertex matrix
of~$E$.

For a modular elliptic curve $E/\mathbb{Q}$, we show a number of
links between the primes $\ell$ for which the mod $\ell$
representation of $E/\mathbb{Q}$ has projective dihedral image and
congruence primes for the newform associated to $E/\mathbb{Q}$.

Let $\mathbf{P}^n$ be the $n$-dimensional projective space over some
algebraically closed field $k$ of characteristic $0$. For an integer
$t\geq 3$ consider the invertible sheaf $O(t)$ on $\mathbf{P}^n$ (Serre
twist of the structure sheaf). Let $N = \binom{t+n}{n}$, the
dimension of the space of global sections of $O(t)$, and let $k$ be an
integer satisfying $0\leq k\leq N - (2n+2)$. Let $P_1,\dots,P_k$
be general points on $\mathbf{P}^n$ and let $\pi \colon X \to
\mathbf{P}^n$ be the blowing-up of $\mathbf{P}^n$ at those points.
Let $E_i = \pi^{-1} (P_i)$ with $1\leq i\leq k$ be the exceptional
divisor. Then $M = \pi^* \bigl( O(t) \bigr) \otimes O_X (-E_1 -
\cdots -E_k)$ is a very ample invertible sheaf on $X$.

On explicite une classe de champ de vecteurs polynomiaux non analytiquement
lin\'earisables \`a l'aide de la correction introduite par \'Ecalle-Vallet.
Notamment, on \'etend des r\'esultats de Schuman sur la trivialit\'e des
hamiltoniens homog\`enes isochrones.

We characterize a class of polynomial vector fields which are not
analytically linearizable using the correction introduced by
\'Ecalle-Vallet. Then, we extend Schuman's result about non
existence of isochronous homogenous Hamiltonian systems.

In this note we show that for an arbitrary reductive Lie group
and any admissible irreducible Banach representation the Mellin
transforms of Whittaker functions extend to meromorphic functions.
We locate the possible poles and show that they always lie along
translates of walls of Weyl chambers.

A partial differential equation, the local M\"obius equation, is
introduced in Riemannian geometry which completely characterizes the
local twisted product structure of a Riemannian manifold. Also the
characterizations of warped product and product structures of
Riemannian manifolds are made by the local M\"obius equation and an
additional partial differential equation.

As a consequence of a theorem of Rost-Springer, we establish that the
cyclicity problem for central simple algebra of degree~5 on fields
containg a fifth root of unity is equivalent to the study of
anisotropic elements of order 5 in the split group of type~$E_8$.

Exact analytical expressions for the inverse Laplace transforms of
the functions $\frac{I_n(s)}{s I_n^\prime(s)}$ are obtained in the
form of trigonometric series. The convergence of the series is
analyzed theoretically, and it is proven that those diverge on an
infinite denumerable set of points. Therefore it is shown that the
inverse transforms have an infinite number of singular points. This
result, to the best of the author's knowledge, is new, as the
inverse transforms of $\frac{I_n(s)}{s I_n^\prime(s)}$ have
previously been considered to be piecewise smooth and continuous.
It is also found that the inverse transforms have an infinite
number of points of finite discontinuity with different left- and
right-side limits. The points of singularity and points of finite
discontinuity alternate, and the sign of the infinity at the
singular points also alternates depending on the order $n$. The
behavior of the inverse transforms in the proximity of the singular
points and the points of finite discontinuity is addressed as well.

For an integer $n \geq 3$, let $M_n$ be the moduli space of spatial polygons
with $n$ edges. We consider the case of odd $n$. Then $M_n$ is a Fano
manifold of complex dimension $n-3$. Let $\Theta_{M_n}$ be the
sheaf of germs of holomorphic sections of the tangent bundle
$TM_n$. In this paper, we prove $H^q (M_n,\Theta_{M_n})=0$ for all
$q \geq 0$ and all odd $n$. In particular, we see that the moduli
space of deformations of the complex structure on $M_n$ consists of
a point. Thus the complex structure on $M_n$ is locally rigid.

We give an upper bound on the essential dimension of the group
$\mathbb{Z}/q\rtimes(\mathbb{Z}/q)^*$ over the rational numbers,
when $q$ is a prime power.

The purpose of this article is to provide criteria for the
simultaneous solvability of the Diophantine equations $X^2 - DY^2 =
c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in
\mathbb{N}$ is not a perfect square. This continues work in
\cite{me}--\cite{alfnme}.

An error in {\it A characterization of left perfect rings},
Canad. Math. Bull. (3) {\bf 38}(1995), 382--384, is indicated
and the consequences identified.

In this paper we study simple associative algebras with finite
$\mathbb{Z}$-gradings. This is done using a simple algebra $F_g$
that has been constructed in Morita theory from a bilinear form
$g\colon U\times V\to A$ over a simple algebra $A$. We show that
finite $\mathbb{Z}$-gradings on $F_g$ are in one to one
correspondence with certain decompositions of the pair $(U,V)$. We
also show that any simple algebra $R$ with finite
$\mathbb{Z}$-grading is graded isomorphic to $F_g$ for some
bilinear from $g\colon U\times V \to A$, where the grading on $F_g$
is determined by a decomposition of $(U,V)$ and the coordinate
algebra $A$ is chosen as a simple ideal of the zero component $R_0$
of $R$. In order to prove these results we first prove similar
results for simple algebras with Peirce gradings.

Langlands has conjectured the existence of a universal group, an
extension of the absolute Galois group, which would play a fundamental
role in the classification of automorphic representations. We shall
describe a possible candidate for this group. We shall also describe
a possible candidate for the complexification of Grothendieck's
motivic Galois group.

A Dirac comb of point measures in Euclidean space with bounded
complex weights that is supported on a lattice $\varGamma$ inherits
certain general properties from the lattice structure. In
particular, its autocorrelation admits a factorization into a
continuous function and the uniform lattice Dirac comb, and its
diffraction measure is periodic, with the dual lattice
$\varGamma^*$ as lattice of periods. This statement remains true
in the setting of a locally compact Abelian group whose topology
has a countable base.

Let $\Phi$ be an algebraically closed field of characteristic zero,
$G$ a finite, not necessarily abelian, group. Given a $G$-grading on
the full matrix algebra $A = M_n(\Phi)$, we decompose $A$ as the
tensor product of graded subalgebras $A = B\otimes C$, $B\cong M_p
(\Phi)$ being a graded division algebra, while the grading of $C\cong
M_q (\Phi)$ is determined by that of the vector space $\Phi^n$. Now
the grading of $A$ is recovered from those of $A$ and $B$ using a
canonical ``induction'' procedure.

We determine the Lie superalgebras that are graded by the root
systems of the basic classical simple Lie superalgebras of type
$C(n)$, $D(m,n)$, $D(2,1;\alpha)$ $(\alpha \in \mathbb{F} \setminus
\{0,-1\})$, $F(4)$, and $G(3)$.

We introduce the notion of Lie algebras with plus-minus pairs as well
as regular plus-minus pairs. These notions deal with certain factorizations
in universal enveloping algebras. We show that many important Lie algebras
have such pairs and we classify, and give a full treatment of, the three
dimensional Lie algebras with plus-minus pairs.

We classify the subalgebras of the general Lie conformal algebra
$\gc_N$ that act irreducibly on $\mathbb{C} [\partial]^N$ and that
are normalized by the sl$_2$-part of a Virasoro element. The
problem turns out to be closely related to classical Jacobi
polynomials $P_n^{(-\sigma,\sigma)}$, $\sigma \in \mathbb{C}$. The
connection goes both ways---we use in our classification some
classical properties of Jacobi polynomials, and we derive from the
theory of conformal algebras some apparently new properties of
Jacobi polynomials.

We begin by reviewing Monstrous Moonshine. The impact of Moonshine on
algebra has been profound, but so far it has had little to teach
number theory. We introduce (using `postcards') a much larger context
in which Monstrous Moonshine naturally sits. This context suggests
Moonshine should indeed have consequences for number theory. We
provide some humble examples of this: new generalisations of Gauss
sums and quadratic reciprocity.

We construct a class of fermions (or bosons) by using a Clifford (or
Weyl) algebra to get two families of irreducible representations for
the extended affine Lie algebra $\widetilde{\mathfrak{gl}_N
(\mathbb{C}_q)}$ of level $(1,0)$ (or $(-1,0)$).

This paper characterizes when a Delone set $X$ in $\mathbb{R}^n$ is an
ideal crystal in terms of restrictions on the number of its local
patches of a given size or on the heterogeneity of their distribution.
For a Delone set $X$, let $N_X (T)$ count the number of
translation-inequivalent patches of radius $T$ in $X$ and let
$M_X(T)$ be the minimum radius such that every closed ball of radius
$M_X(T)$ contains the center of a patch of every one of these kinds.
We show that for each of these functions there is a
``gap in the spectrum'' of possible growth rates between being
bounded and having linear growth, and that having sufficiently
slow linear growth is equivalent to $X$ being an ideal crystal.

Explicitly, for $N_X(T)$, if $R$ is the covering radius of $X$
then either $N_X(T)$ is bounded or $N_X (T) \ge T/2R$ for all $T>0$.
The constant $1/2R$ in this bound is best possible in all dimensions.
For $M_X(T)$, either $M_X(T)$ is bounded or $M_X(T)\ge T/3$ for all $T>0$.
Examples show that the constant $1/3$ in this bound cannot be replaced by
any number exceeding $1/2$. We also show that every aperiodic Delone
set $X$ has $M_X(T)\ge c(n) T$ for all $T>0$, for a certain constant $c(n)$
which depends on the dimension $n$ of $X$ and is $>1/3$ when $n>1$.

We study the representations of extended affine Lie algebras
$s\ell_{\ell+1} (\mathbb{C}_q)$ where $q$ is $N$-th primitive root of
unity ($\mathbb{C}_q$ is the quantum torus in two variables). We
first prove that $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ for a
suitable number of copies is a quotient of $s\ell_{\ell+1}
(\mathbb{C}_q)$. Thus any finite dimensional irreducible module for
$\bigoplus s\ell_{\ell+1} (\mathbb{C})$ lifts to a representation of
$s\ell_{\ell+1} (\mathbb{C}_q)$. Conversely, we prove that any finite
dimensional irreducible module for $s\ell_{\ell+1} (\mathbb{C}_q)$
comes from above. We then construct modules for the extended affine
Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q) \oplus \mathbb{C} d_1
\oplus \mathbb{C} d_2$ which is integrable and has finite dimensional
weight spaces.

We embed the moduli space $Q$ of 5 points on the projective line
$S_5$-equivariantly into $\mathbb{P} (V)$, where $V$ is the
6-dimensional irreducible module of the symmetric group $S_5$. This
module splits with respect to the icosahedral group $A_5$ into the two
standard 3-dimensional representations. The resulting linear
projections of $Q$ relate the action of $A_5$ on $Q$ to those on the
regular icosahedron.

We consider two dynamical systems associated with a substitution of
Pisot type: the usual $\mathbb{Z}$-action on a sequence space, and
the $\mathbb{R}$-action, which can be defined as a tiling dynamical
system or as a suspension flow. We describe procedures for checking
when these systems have pure discrete spectrum (the ``balanced
pairs algorithm'' and the ``overlap algorithm'') and study the
relation between them. In particular, we show that pure discrete
spectrum for the $\mathbb{R}$-action implies pure discrete spectrum
for the $\mathbb{Z}$-action, and obtain a partial result in the
other direction. As a corollary, we prove pure discrete spectrum
for every $\mathbb{R}$-action associated with a two-symbol
substitution of Pisot type (this is conjectured for an arbitrary
number of symbols).

Quantum tori with graded involution appear as coordinate algebras of
extended affine Lie algebras of type $\rmA_1$, $\rmC$ and $\BC$.
We classify them in the category of algebras with involution. From
this, we obtain precise information on the root systems of extended
affine Lie algebras of type $\rmC$.