For $K$ a connected finite complex and $G$ a compact connected Lie group, a finiteness result is proved for gauge groups ${\mathcal G}(P)$ of principal $G$-bundles $P$ over $K$: as $P$ ranges over all principal $G$-bundles with base $K$, the number of homotopy types of ${\mathcal G}(P)$ is finite; indeed this remains true when these gauge groups are classified by $H$-equivalence, that is, homotopy equivalences which respect multiplication up to homotopy. A case study is given for $K = S^4$, $G = \text{SU}(2)$: there are eighteen $H$-equivalence classes of gauge group in this case. These questions are studied via fibre homotopy theory of bundles of groups; the calculations in the case study involve $K$-theories and $e$-invariants. 1991 Mathematics Subject Classification: 54C35, 55P15, 55R10.

Let ${\cal M} =\langle M,<,\ldots\rangle$ be alinearly ordered structure. We define ${\cal M}$ to be{\em o-minimal} if every definable subset of $M$ is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo-minimal ${\cal M}$ can induce on a neighbourhood of any $a$in $M$. Roughly said, one of the following holds: \begin{enumerate}\item[(i)] $a$ is trivial (technical term), {\em or}\item[(ii)] $a$ has a convex neighbourhood on which ${\cal M}$ induces the structure of an ordered vector space, {\em or}\item[(iii)] $a$ is contained in an open interval on which ${\cal M}$ induces the structure of an expansion of a real closed field. \end{enumerate} The proof uses ‘geometric calculus’ which allows one to recover a differentiable structure by purely geometric methods.

1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.

We prove the quantum version - for Hecke algebras $H(A_n)$ of type $A$ at roots of unity - of Kleshchev's modular branching rule for symmetric groups. This result describes the socle of the restriction of an irreducible $H(A_n)$-module to the subalgebra $H(A_{n-1})$. As a consequence, we describe the quantum version of the Mullineux involution describing the irreducible module obtained on twisting an irreducible module with the sign representation.

1991 Mathematics Subject Classification: 20C05, 20G05.

Bihermitian complex surfaces are oriented conformal four-manifolds admitting two independent compatible complex structures. Non-anti-self-dual bihermitian structures on ${\mathbb R}^4$ and the four-dimensional torus $T^4$ have recently been discovered by P. Kobak. We show that an oriented compact 4-manifold, admitting a non-anti-self-dual bihermitian structure, is a torus or K3 surface in the strongly bihermitian case (when the two complex structures are independent at each point) or, otherwise, must be obtained from the complex projective plane or a minimal ruled surface of genus less than 2 by blowing up points along some anti-canonical divisor (but the actual existence of bihermitian structures in the latter case is still an open question). The paper includes a general method for constructing non-anti-self-dual bihermitian structures on tori, K3 surfaces and $S^1\times S^3$. Further properties of compact bihermitian surfaces are also investigated.

1991 Mathematics Subject Classification: 53C12, 53C55, 32J15.

In this paper, we develop the crystal basis theory for quantum generalized Kac–Moody algebras. For a quantum generalized Kac–Moody algebra $U_q(\mathfrak{g})$, we first introduce the category $\mathcal{O}_{int}$ of $U_q(\mathfrak{g})$-modules and prove its semisimplicity. Next, we define the notion of crystal bases for $U_q(\mathfrak{g})$-modules in the category $\mathcal{O}_{int}$ and for the subalgebra $U_q^-(\mathfrak{g})$. We then prove the tensor product rule and the existence theorem for crystal bases. Finally, we construct the global bases for $U_q(\mathfrak{g})$-modules in the category $\mathcal{O}_{int}$ and for the subalgebra $U_q^-(\mathfrak{g})$.

Green's function $G(x)$ of a zero mean random walk on the $N$-dimensional integer lattice ($N\geq 2$) is expanded in powers of $1/|x|$ under suitable moment conditions. In particular, we find minimal moment conditions for $G(x)$ to behave like a constant times the Newtonian potential (or logarithmic potential in two dimensions) for large values of $|x|$. Asymptotic estimates of $G(x)$ in dimensions $N\geq 4$, which are valid even when these moment conditions are violated, are computed. Such estimates are applied to determine the Martin boundary of the random walk. If $N= 3$ or $4$ and the random walk has zero mean and finite second moment,the Martin boundary consists of one point, whereas if $N\geq 5$, this is not the case, because non-harmonic functions arise as Martin boundary points for a large class of such random walks. A criterion for when this happens is provided.

1991 Mathematics Subject Classification: 60J15, 60J45, 31C20.

If $G$ is a pro-$p$, $p$-adic, Lie group containing no element of order $p$ and if $\Lambda (G)$ denotes the Iwasawa algebra of $G$ then we propose a number of invariants associated to finitely generated $\Lambda (G)$-modules, all given by various forms of Euler characteristic. The first turns out to be none other than the rank, and this gives a particularly convenient way of calculating the rank of Iwasawa modules. Others seem to play similar roles to the classical Iwasawa $\lambda $- and $\mu $-invariants. We explore some properties and give applications to the Iwasawa theory of elliptic curves.

2000 Mathematical Subject Classification: primary 16E10; seconday 11R23.

Let $A = (a^{ij})$ be a Borel mapping on $[0, 1] \times \mathbb{R}^d$ with values in the space of non-negative operators on $\mathbb{R}^d$ and let $b = (b^i)$ be a Borel mapping on $[0, 1] \times \mathbb{R}^d$ with values in $\mathbb{R}^d$. Let

\[Lu(t, x) = \partial_{t} u(t, x) + a^{ij}(t, x)\partial_{x_i}\partial_{x_j}u(t, x) + b^i(t, x)\partial_{x_i}u(t, x), \quad u \in C_0^\infty((0, 1)\times \mathbb{R}^d).\]

Under broad assumptions on $A$ and $b$, we construct a family $\mu = (\mu_t)_{t \in [0, 1)}$ of probability measures $\mu_t$ on $\mathbb{R}^d$ which solves the Cauchy problem $L^{*}\mu = 0$ with initial condition $\mu_0 = \nu$, where $\nu$ is a probability measure on $\mathbb{R}^d$, in the following weak sense:

$$\[ \int_0^1\int_{\mathbb{R}^d} Lu(t, x)\, \mu_t(dx)\, dt=0, \quad u \in C_0^\infty((0, 1) \times \mathbb{R}^d), \] $$

and

$$\[ \lim\limits_{t \to 0} \int_{\mathbb{R}^d} \zeta(x)\, \mu_t(dx) = \int_{\mathbb{R}^d} \zeta(x)\, \nu(dx), \quad \zeta \in C_0^\infty(\mathbb{R}^d). \] $$

Such an equation is satisfied by transition probabilities of a diffusion process associated with $A$ and $b$ provided such a process exists. However, we do not assume the existence of a process and allow quite singular coefficients, in particular, $b$ may be locally unbounded or $A$ may be degenerate. An infinite-dimensional analogue is discussed as well. Main methods are $L^p$-analysis with respect to suitably chosen measures and reduction to the elliptic case (studied previously) by piecewise constant approximations in time.

In this paper we find the multiplicities $\dim L(\lambda)_{\lambda-\alpha}$ where $\alpha$ is an {\em arbitrary} root and $L(\lambda)$ is an irreducible $SL_n$-module with highest weight $\lambda$. We provide different bases of the corresponding weight spaces and outline some applications to the symmetric groups. In particular we describe certain composition multiplicities in the modular branching rule.

1991 Mathematics Subject Classification: 20C05, 20G05.

In 1959 E. Bishop used a Banach-space version of the analytic duality principle established by e Silva, K\"{o}the, Grothendieck and others to study connections between spectral decomposition properties of a Banach-space operator and its adjoint. According to Bishop a continuous linear operator $T \in L(X)$ on a Banach space $X$ satisfies property $(\beta)$ if the multiplication operator ${\cal O}(U,X) \rightarrow {\cal O}(U,X),$ $f \mapsto (z-T)f,$ is injective with closed range for each open set $U$ in the complex plane. In the present article the analytic duality principle in its original locally convex form is used to develop a complete duality theory for property $(\beta)$. At the same time it is shown that, up to similarity, property $(\beta)$ characterizes those operators occurring as restrictions of operators decomposable in the sense of C. Foias, and that its dual property, formulated as a spectral decomposition property for the spectral subspaces of the given operator, characterizes those operators occurring as quotients of decomposable operators. It is proved that, unlike the situation for commuting subnormal operators, each finite commuting system of operators with property $(\beta)$ can be extended to a finite commuting system of decomposable operators. Meanwhile the results of this paper have been used to prove the existence of invariant subspaces for subdecomposable operators with sufficiently rich spectrum.

1991 Mathematics Subject Classification: 47A11, 47B40.

We study cubical sets without degeneracies, which we call $\square$-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a $\square$-set $C$ has an infinite family of associated $\square$-sets $J^i(C)$, for $i=1,2,\ldots$, which we call James complexes. There are mock bundle projections $p_i \colon |J^i (C)| \to |C|$ (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of $\Omega (S^2)$. The algebra of these classes mimics the algebra of the cohomotopy of $\Omega (S^2)$ and the reduction to cohomology defines a sequence of natural characteristic classes for a $\square$-set. An associated map to $BO$ leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation.

Given weight functions $\theta$, $w$, $\rho$ and $v$, the weighted modular inequality $$ Q^{-1}\bigg(\int_{0}^{\infty} Q(\theta(x)Tf(x)) w(x)\, dx\bigg) \leq P^{-1}\bigg(\int_{0}^{\infty} P(C \rho(x) f(x)) v(x)\, dx\bigg)$$ is characterized. Here $Q$ is a strictly increasing function with $Q(0) = 0$, $Q(\infty) = \infty$ and $2Q(x) \leq Q(C x)$, $P$ is a Young's function, and $T$ is the Hardy operator or a Hardy type operator. In particular, a characterizing condition for the Hardy type operator to map $L^{p}(w)$ to $L^{q}(v)$ when $0 < q < 1 \leq p < \infty$ is deduced. In addition, a new proof for the Maz'ja-Sinnamon theorem is given, and weighted Lorentz norm inequalities for Hardy type operators are established.

1991 Mathematics Subject Classification: primary 26D15, 42B25; secondary 26A33, 46E30.

We generalize the idea of reflexibility of a map on a surface by introducing certain integers as its ”exponents“. An exponent is any integer $e$ with the property that changing the cyclic permutation of edges around each vertex induced by the map to its $e$-th power gives rise to an isomorphic map. The exponents reduced modulo the least common multiple of the vertex valencies form an Abelian group, the exponent group $\mbox{Ex}\,(M)$ of the map $M$. Along with the automorphism group, the group in fact provides an additional measure of symmetry of ~$M$.

The paper is devoted to developing the fundamentals of the theory of exponent groups of maps. Motivation comes from the problem of classification of regular maps with a given underlying graph. To this end, we prove that the number of non-isomorphic regular maps (if any) with a given underlying graph and the same map automorphism group is $|{\Bbb Z}_n^*:\mbox{Ex}\,(M)|$, $n$ being the valency of $M$. We calculate the exponent groups for some interesting families of regular maps including complete maps. Special attention is paid to the problem of how the antipodality of a map is reflected by its exponent group. In the final section we discuss several open problems.

1991 Mathematics Subject Classification: 05C10, 05C25, 20F32.

Let $\Gamma$ be a countable abelian semigroup and $\mathcal{A}$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(\Gamma \times \mathcal{A})$-graded Lie superalgebra $\mathfrak{L} =\bigoplus_{(\alpha, a) \Gamma \times \mathcal{A}} \mathfrak{L}_{(\alpha, a)}$ by Lie superalgebra automorphisms preserving the $(\Gamma \times \mathcal{A})$-gradation. In this paper, we show that the Euler--Poincar\'e principle yields the generalized denominator identity for $\mathfrak{L}$ and derive a closed form formula for the supertraces $\text{str}(g| \mathcal{L}_{(\alpha, a)})$ for all $g\in G$, where $(\alpha, a) \in \Gamma \times \mathcal{A}$. We discuss the applications of our supertrace formula to various classes of infinite-dimensional Lie superalgebras such as free Lie superalgebras and generalized Kac--Moody superalgebras. In particular, we determine the decomposition of free Lie superalgebras into a direct sum of irreducible $\text{GL}(n) \times \text{GL}(k)$-modules, and the supertraces of the Monstrous Lie superalgebras with group actions. Finally, we prove that the generalized characters of Verma modules and irreducible highest-weight modules over a generalized Kac--Moody superalgebra $\mathfrak{g}$ corresponding to the Dynkin diagram automorphism $\sigma$ are the same as the usual characters of Verma modules and irreducible highest-weight modules over the orbit Lie superalgebra $\breve{\mathfrak{g}} = \mathfrak{g}(\sigma)$ determined by $\sigma$. 1991 Mathematics Subject Classification: 17A70, 17B01, 17B65, 17B70, 11F22.

This paper is concerned with rational representations of reductive algebraic groups over fields of positive characteristic $p$. Let $G$ be a simple algebraic group of rank $\ell$. It is shown that a rational representation of $G$ is semisimple provided that its dimension does not exceed $\ell p$. Furthermore, this result is improved by introducing a certain quantity $\mathcal{C}$ which is a quadratic function of $\ell$. Roughly speaking, it is shown that any rational $G$ module of dimension less than $\mathcal{C} p$ is either semisimple or involves a subquotient from a finite list of exceptional modules.

Suppose that $L_1$ and $L_2$ are irreducible representations of $G$. The essential problem is to study the possible extensions between $L_1$ and $L_2$ provided $\dim L_1 + \dim L_2$ is smaller than $\mathcal{C} p$. In this paper, all relevant simple modules $L_i$ are characterized, the restricted Lie algebra cohomology with coefficients in $L_i$ is determined, and the decomposition of the corresponding Weyl modules is analysed. These data are then exploited to obtain the needed control of the extension theory.

1991 Mathematics Subject Classification: 20G05.

In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number $\xi$ by algebraic integers of degree at most 3. They did so, using geometry of numbers, by resorting to the dual problem of finding simultaneous approximations to $\xi$ and $\xi^2$ by rational numbers with the same denominator. In this paper, we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers $\xi$. We give several properties of these numbers including measures of approximation by rational numbers, by quadratic real numbers and by algebraic integers of degree at most 3.

We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory.

2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)

Given any sequence of non-abelian finite simple primitive permutation groups $(S_{n})$, we construct a finitely generated group $G$ whose profinite completion is the infinite permutational wreath product $\ldots S_{n}\wr S_{n-1}\wr\ldots\wr S_{0}$. It follows that the upper composition factors of $G$ are exactly the groups $S_{n}$. By suitably choosing the sequence $(S_{n})$ we can arrange that $G$ has any one of a continuous range of slow, non-polynomial subgroup growth types. We also construct a $61$-generator perfect group that has every non-abelian finite simple group as a quotient. 2000 Mathematics Subject Classification: 20E07, 20E08, 20E18, 20E32.

Let $V$ denote the classical Volterra operator. In this work, sharp estimates of the norm of $(I - V)^n$ acting on $L^p [0, 1]$, for $1 \leq p \leq \infty$, are obtained. As a consequence, $I - V$ acting on $L^p [0, 1]$, with $1 \leq p \leq \infty$, is power bounded if and only if $p = 2$. Thus the Volterra operator characterizes when $L^p [0, 1]$ is a Hilbert space. By means of sharp estimates of the $L^1$-norm of the $n$th partial sums of the generating function of the Laguerre polynomials on the unit circle, it is also proved that <formula form="inline" disc="math" id="frm012"><formtex notation="AMSTeX">$I - V$ is uniformly Kreiss bounded on the spaces $L^p [0,1]$, for $1 \leq p \leq \infty$.

A bounded linear operator $T$ on a Banach space is said to be Kreiss bounded if there is a constant $C > 0$ such that $\Vert (T - \lambda)^{-1} \Vert \leq C( | \lambda | - 1)^{-1}$ for $| \lambda | > 1$. If the same upper estimate holds for each of the partial sums of the resolvent, then $T$ is said to be uniformly Kreiss bounded. This is, for instance, true for power bounded operators. For finite-dimensional Banach spaces, Kreiss' Matrix Theorem asserts that Kreiss boundedness is equivalent to $T$ being power bounded. Thus, in the infinite-dimensional setting, even a much stronger property than Kreiss boundedness still does not imply power boundedness. It is also shown that, for general operators, uniform Abel boundedness characterizes Cesàro boundedness and, as a consequence, uniform Kreiss boundedness is characterized in terms of a Cesàro type boundedness of order 1.

The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah and Bott asserts that its rational cohomology ring is generated by the universal classes, that is, by the Künneth components of the Chern classes of the universal bundle.

This paper studies the larger, non-compact moduli space of Higgs bundles, as introduced by Hitchin and Simpson, with values in the canonical bundle $K$. This is diffeomorphic to the space of all connections of central constant curvature, whether unitary or not. The main result of the paper is that, in the rank 2 case, the rational cohomology ring of this space is again generated by universal classes.

The spaces of Higgs bundles with values in $K(n)$ for $n > 0$ turn out to be essential to the story. Indeed, we show that their direct limit has the homotopy type of the classifying space of the gauge group, and hence has cohomology generated by universal classes.