A construction is described of a 153-dimensional representation of the triple cover of O'Nan's sporadic simple group, over the field of order 4. This is then extended to a 306-dimensional representation of 3·O'N[ratio ]2 over GF(2). A similar construction in characteristic 3 gives an unrelated representation of the O'Nan group in 154 dimensions over GF(3), which extends to O'N[ratio ]2 provided that the field is extended to GF(9).

Projections are constructed in the rotation algebra that are orthogonal to their Fourier transform and which are fixed under the flip automorphism. Such projections are expected in a construction of an inductive limit structure for the irrational rotation algebra that is invariant under the Fourier transform (namely, as two circle algebras of the same dimension, which are swapped by the Fourier transform, plus a few points). The calculation is based on Rieffel's construction of the Schwartz space as an equivalence bimodule of rotation algebras as well as on the theory of theta functions.

Symplectic groups are well known as the groups of isometries of a vector space with a non-singular bilinear alternating form. These notions can be extended by replacing the vector space by a module over a ring R, but if R is non-commutative, it will also have to have an involution. We shall here be concerned with symplectic groups over free associative algebras (with a suitably defined involution). It is known that the general linear group GLn over the free algebra is generated by the set of all elementary and diagonal matrices (see [1, Proposition 2.8.2, p. 124]). Our object here is to prove that the symplectic group over the free algebra is generated by the set of all elementary symplectic matrices. For the lowest order this result was obtained in [4]; the general case is rather more involved. It makes use of the notion of transduction (see [1, 2.4, p. 105]). When there is only a single variable over a field, the free algebra reduces to the polynomial ring and the weak algorithm becomes the familiar division algorithm. In that case the result has been proved in [3, Anhang 5].

The following result is well known and easy to prove (see [14, Theorem 2.2.6]).

Theorem 0. If A is a primitive associative Banach algebra, then there exists a Banach space X such that A can be seen as a subalgebra of the Banach algebra BL(X) of all bounded linear operators on X in such a way that A acts irreducibly on X and the inclusion A[rarrhk ]BL(X) is continuous.

In fact, if X is any vector space on which the primitive Banach algebra A acts faithfully and irreducibly, then X can be converted in a Banach space in such a way that the requirements in Theorem 0 are satisfied and even the inclusion A[rarrhk ]BL(X) is contractive.

Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of Theorem 0.

The following theorem is proved: if $(M^{4n+1},g)$ is a complete Riemannian manifold and $\Sigma\subset M$ is an oriented hypersurface partitioning $M$ and with non-zero signature, then the spectrum of the Hodge–deRham Laplacian is $[0,\infty[$ . This result is obtained by a new Callias-type index. This new formula links half-bound harmonic forms (that is, nearly $L^2$ but not in $L^2$ ) with the signature of $\Sigma$ .

Motivated by, but independent of, some recent work in quantum stochastic calculus, a theory of differential and integral calculus is developed which is intrinsic to the universal enveloping algebra of a Lie algebra whose Lie bracket is obtained by taking commutators in an associative algebra. The differential map satisfies a generalisation of Leibniz' formula called the Leibniz–Itô formula, which involves the associative multiplication. There is an analogue of the Taylor–Maclaurin expansion. Through passing to formal power series, a theory of product integrals is developed; such integrals are characterised by a group-like property with respect to the coproduct.

A real algebraic, plane, projective curve $A$ of degree $m$ is given by a homogeneous polynomial of degree $m$ in three variables, with real coefficients, defined up to multiplication by a non-zero scalar. If $F$ is such a polynomial defining $A$ , we denote by ${\bb C} A$ and ${\bb R} A$ , respectively, the sets of solutions of the equation $F = 0$ in ${\bb C} P^2$ , respectively ${\bb R} P^2$ . We suppose that the curve $A$ is non-singular, that is, $F$ has no critical points in ${\bb C}^3\backslash 0$ . Then ${\bb C} A$ is a Riemannian surface of genus $g = (m-1)(m-2)/2$ , and ${\bb R} A$ is a collection of $L \le g+1$ circles embedded in ${\bb R} P^2$ . If $L = g+1$ , we say that $A$ is an $M$ -curve. A circle embedded in ${\bb R} P^2$ is called oval, or pseudo-line, depending on whether it realizes the class 0 or 1 of $H_1({\bb R} P^2)$ . If $m$ is even, the $L$ components of ${\bb R} A$ are ovals; if $m$ is odd, ${\bb R} A$ contains exactly one pseudo-line, which will be denoted by ${\cal J}$ . Note that an oval separates ${\bb R} P^2$ into two pieces, a Möbius band and a disc. The latter is called the interior of the oval. An oval of ${\bb R} A$ is said to be empty if its interior contains no other oval of ${\bb R} A$ . Two ovals form an injective pair if one of them lies in the interior of the other one.

We give a necessary and sufficient condition on an elliptic symbol σ of order m to ensure that the unique closed extension in Lp(ℝn) for 1 < p < ∞, of the pseudo-differential operator Tσ, initially defined on the Schwartz space, is a Fredholm operator from Lp(ℝn) into Lp(ℝn) with domain Hm, p, where Hm, p is the Lp Sobolev space of order m.

The classification of finite groups acting freely on a homology 3-sphere is still incomplete. However it is known that these groups have periodic cohomology and groups with periodic cohomology are classified by the Suzuki–Zassenhaus theorem. The paper considers, more generally, finite groups which may act on a homology 3-sphere possibly with fixed points and the natural generalization of the Suzuki–Zassenhaus theorem to this case is found. According to the Suzuki–Zassenhaus theorem a group with periodic cohomology is either solvable or has a unique composition factor which is not cyclic, isomorphic to PSL(2, q) for q an odd prime. The main point of the work is that this fact is still true in the more general situation.

The paper continues the investigation of the Hecke structure of spaces of half-integral weight cusp forms ${S_{k + 1/2}}(4N, \chi)$, where $k$ and $N$ are positive integers with $N$ odd, and $\chi$ is an even quadratic Dirichlet character modulo $4N$. In the Hecke decomposition of these spaces, contributions are determined arising from newforms which are quadratic twists of newforms at lower levels. Combining this result with its counterpart in a previous paper regarding non-twists gives this paper's main result: necessary and sufficient conditions under which a given newform has equivalent cusp forms in ${S_{k + 1/2}}(4N, \chi)$. The result reformulates, in classical terms, the representation-theoretic conditions given by Flicker and Waldspurger. The conditions involve easily-verified information about the primes dividing the level of the newform, and about the behavior of the newform under certain quadratic twists and Atkin–Lehner involutions. The theorem is applied to give explicit examples of twisted newforms having no equivalent half-integral weight cusp forms in any space ${S_{k + 1/2}}(4N, \chi)$ as above.

The paper proves the existence and multiplicity of periodic solutions of the equation

formula here

where g[ratio ]R→R is of class C1, h[ratio ]R→R is continuous and 2π-periodic, and s is a parameter.

It is shown that the natural action of the absolute Galois group on the ideals defining principal congruence subgroups of certain nonarithmetic Fuchsian triangle groups is compatible with its action on the algebraic curves that these congruence groups uniformize.

Let $U^+$ be the plus part of the quantized enveloping algebra of a simple Lie algebra of type $A_n$ and let $\mathcal{B}^*$ be the dual canonical basis of $U^+$. Let $b$, $b'$ be in $\mathcal{B}^*$, and suppose that one of the two elements is a $q$-commuting product of quantum flag minors. It is shown that $b$ and $b'$ are multiplicative if and only if they $q$-commute.

Scharlemann and Thompson introduced an invariant $\rho(K,\gamma)\in \mathbb{Q}/2\mathbb{Z}$ for the pair of a knot $K$ in the 3-sphere and an unknotting tunnel $\gamma$ for $K$. The paper studies the relationship between the invariant $\rho(K,\gamma)$ of a (1,1)-knot and the distance of its (1,1)-splitting introduced by the author.

Let $A > 0$ be an integer. The equation $x^5 - y^5 = Az^5$ was first studied by Dirichlet and Lebesgue. Lebesgue conjectured in 1843 that if $A$ has no prime divisors of the form $10k+1$, the equation has no solutions except the visible ones. Partial results were obtained by Lebesgue and by Terjanian in 1987. The purpose of the paper is to prove Lebesgue's conjecture. The main tool used is the method known as the elliptic Chabauty method.

Let $\Omega X$ be the space of Moore loops on a finite, $q$-connected, $n$-dimensional CW complex $X$, and let $R\subset\Q$ be a subring containing 1/2. Let $\rho\left(R\right)$ be the least non-invertible prime in $R$. For a graded $R$-module $M$ of finite type, let $FM = M / {\rm Torsion}\,M$. We show that the inclusion $P \subset FH_{*}\left(\Omega X;R\right)$ of the sub-Lie algebra of primitive elements induces an isomorphism of Hopf algebras $$UP \overset{\cong}{\longrightarrow} FH_{*}\left(\Omega X;R\right)},$$ provided that $\rho\left(R\right) \geqslant n/q$. Furthermore, the Hurewicz homomorphism induces an embedding of $F(\pi_{*}\left(\Omega X\right)\otimes R)$ in $P$, with $P/F(\pi_{*}\left(\Omega X\right)\otimes R)$ torsion. As a corollary, if $X$ is elliptic, then $FH_{*}\left(\Omega X;R\right)$ is a finitely generated $R$-algebra.

A relation is described between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M. Kobayashi. It is shown that this relation continues to hold for the extension of Arnold's strange duality found by C. T. C. Wall and the author. By a method of Ehlers–Varchenko, the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. This method is generalized to the isolated complete intersection singularities embraced in the extended duality. This is used to explain the duality of characteristic polynomials of the monodromy discovered by K. Saito for Arnold's original strange duality and extended by the author to the other cases.

For a prime p, a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod-p homology. Associated to a modular representation G → Gl(n; [ ]p), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so-called subgroup, centralizer and normalizer decompositions. For an action of G on an [ ]p-vector space V, this collection consists of all subgroups of G with nontrivial p-Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG.

One of the principal impulses for this paper was the consideration of the question, first raised in 1973 by G. Baumslag [2], of whether every finitely generated Abelian by polycyclic group can be embedded in a finitely presented Abelian by polycyclic group. That this cannot be done follows at once from

THEOREM A. Every finitely presented Abelian by polycyclic group has a metanilpotent normal subgroup of finite index.

Two classes of vector-valued BMOA spaces are defined, in the complex ball and on the complex sphere, respectively. In the case of the complex sphere, vector measures are involved, since the argument in the scalar setting is not appropriate. Several properties (the Lp-equivalent norm theorem, exponential decay, the Baernstein theorem, and so on) of BMOA in the complex ball are extended to the Banach space setting. The two classes of BMOA spaces are proved to be isomorphic; in particular, the corresponding John–Nirenberg exponential decay is shown. Finally, the vector-valued H1-BMOA duality theorem is proved.