Let $G$ be a simple simply connected complex Lie group. Some criteria are given for the nonexistence of exceptional principal $G$-bundles over a complex projective surface. As an application, it is shown that there are no exceptional $G$-bundles over a surface whose arithmetic genus is zero or one. It is also shown that there are no stable exceptional $G$-bundles over an abelian surface.

The concept of Morita equivalence is generalized to the context of locally C$^{*}$-algebras. This generalizes a well-known theorem of Brown, Green and Rieffel, Pacific J. Math. 71 (1977) 349–363.

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane that can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in ${\rm SL}(2,{\mathbb Z})$. It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. An example of a holomorphic almost modular form is the logarithm of $\prod_{n=1}^\infty (1-\exp(2\pi\i n^2 z))$. The paper is motivated by the author's previous studies [Int. Math. Res. Not. 39 (2003) 2131–2151] on the connection between almost modular functions and the distribution of the sequence $n^2x$ modulo one.

In this paper, perturbations of the left and right essential spectra of $2\times 2$ upper triangular operator matrix $M_C$ are studied, where $M_C= \left(\begin{smallmatrix} A & C 0 & B \end{smallmatrix}\right)$ is an operator acting on the Hilbert space ${\cal H}\oplus{\cal K }$. For given operators $A$ and $B$, the sets $\bigcap_{C\in B({\cal K},{\cal H})}\sigma_{\rle}(M_C)$ and $\bigcap_{C\in B({\cal K},{\cal H})}\sigma_{\rre}(M_C)$ are determined, where $ \sigma_{\rle}(T)$ and $\sigma_{\rre}(T)$ denote, respectively, the left essential spectrum and the right essential spectrum of an operator $T$.

By using a more efficient way of ‘almost diagonalizing’, known results on small fractional parts of general real cubic forms in a small number of variables, are improved. In particular, for ternary forms the method gives a bound superior to that arising from simply reducing to a cubic form in one variable.

The semilinear elliptic eigenvalue problem with superlinear pure power nonlinearity is considered. This problem is treated from the standpoint of $L^2$-theory and the precise asymptotic formula for the eigenvalue parameter $\lambda \,{=}\, \lambda(\alpha)$ as $\alpha \,{\to}\, \infty$ is established, where $\alpha$ is the $L^2$-norm of the solution $u$ associated with $\lambda$.

An inequality is established involving colengths of the tight closure of ideals of systems of parameters in local rings with some mild conditions. As an application, a proof is given of a result due to Goto and Nakamura (first conjectured by Watanabe and Yoshida), which states that the Hilbert–Samuel multiplicity of a parameter ideal is greater than or equal to the colength of the tight closure of the ideal. The result is also further refined.

The purpose of this paper is to show how the methods of motivic integration of Kontsevich, Denef–Loeser (Invent. Math. 135 (1999) 201–232 and Compositio Math. 131 (2002) 267–290) and Looijenga (Astérisque 276 (2002) 267–297) can be adapted to prove the McKay–Ruan correspondence, a generalization of the McKay–Reid correspondence to orbifolds that are not necessarily global quotients.

Consider the family of exponential maps $\Ek(z)=\exp(z)+\kappa$. This paper shows that any unbounded Siegel disk $U$ of $\Ek$ contains the singular value $\kappa$ on its boundary. By a result of Herman, this implies that $\kappa\in \partial U$ if the rotation number is diophantine.

Let $G\subseteq \mathbb{C}^{2}$ be the open symmetrized bidisc, namely $G= \{(\lambda_{1}+\lambda_{2},\lambda_{1}\lambda_{2}):|\lambda_{1}|<1,|\lambda_{2}|<1\}$. In this paper, a proof is given that $G$ is not biholomorphic to any convex domain in $\mathbb{C}^{2}$. By combining this result with earlier work of Agler and Young, the author shows that $G$ is a bounded domain on which the Carathéodory distance and the Kobayashi distance coincide, but which is not biholomorphic to a convex set.

Two normalised Hecke eigenforms $f$ and $g$ are considered in this paper, of level one and weights $k'\,{>}\,k$, lying in a $p$-adic family, such that $f\,{\equiv}\,g\pmod{p^r}$ as $q$-expansions. Interpreting the congruence in terms of the associated Galois representations leads to the existence of non-trivial global $p^r$-torsion for the motive associated with the tensor product $L$-function at $s\,{=}\,k'\,{-}\,1$. (It must be assumed that the (mod $p$) Galois representation attached to $g$ is irreducible.) This contributes a factor of $p^r$ to the denominator of the Bloch–Kato conjectural formula for the $L$-value. The $p$-part of the numerator is considered, using recent work of Diamond, Flach and Guo. Using Shimura's Rankin–Selberg integral formula and the Clausen–von Staudt theorem, the ratio of $L$-values is examined at $s\,{=}\,k'-1$, for the tensor product and a quadratic twist; confirmation is given (under certain conditions) that, at $p$, this is as predicted by Bloch and Kato.

A well-known formula of Bendixson states that solutions of first-order differential equations, as functions of their initial conditions, satisfy a certain partial differential equation. A consequence is Alekseev's nonlinear variation of parameters formula. In this paper, corresponding results are proved for difference equations. To achieve this, use is made of the recently introduced concept of alpha derivatives, rather than of differences or of the usual derivatives. This technique allows the results to be generalized to alpha dynamic equations, which include among others ordinary differential and difference equations.

Several inequalities for trace norms of sums of $n$ operators with roots of unity coefficients are proved in this paper. When $n=2$, these reduce to the classical Clarkson inequalities and their non-commutative analogues.

Let ${\cal A}$ be a unital maximal full algebra of operator fields with base space $[0, 1]^k$ and fibre algebras $\{{\cal A}_t\}_{t\in[0, 1]}^{k}$. It is shown in this paper that the stable rank of ${\cal A}$ is bounded above by the quantity sup$_{t\in[0, 1]^k}\,{\rm sr}(C([0, 1]^k)\,{\otimes}\,{\cal A}_t)$, where ‘sr’ means stable rank. Using the above estimate, the stable ranks of the C$^*$-algebras of the (possibly higher rank) discrete Heisenberg groups are computed.

Let $S$ be a compact Riemann surface of genus ${\rm g}\,{>}\,1$. By using Teichmüller theory, a new approach is obtained to the existence of a Jenkins–Strebel differential with prescribed type and heights on $S$. Moreover, the geometric structures of horizontal trajectories of certain classes of quadratic differentials are discussed.

Let $G$ be a finitely generated, discrete group of polynomial growth. This paper provides a proof of the boundedness in $L_p(G)$, $1<p<\infty$, of some higher-order Riesz transform operators associated with a probability density on $G$.

There is a Weil cohomology theory with coefficients in a pseudofinite field of characteristic zero.

The omega limit sets of a nonlinear operator $T$ which is defined on a positive cone and satisfies certain ray-contractive type conditions are discussed. Under the assumption that the restriction of $T$ to a compact subset is surjective, the following alternatives are proved: the omega limit set of a point in the cone either consists of a fixed point or forms a 2-cycle. In addition, new proofs and extensions to relevant results are given.

It is shown that, given continuous, absolutely irreducible representations of ${\rm Gal}(\mathbb{Q}^{\rm ac}/\mathbb{Q})$ with values in ${\rm GL}_2(\mathbb{F}_2)$ and ${\rm GL}_2(\mathbb{F}_3)$, and having cyclotomic determinant, there is a weight 2 newform of some level whose mod $2$ and mod $3$ representations are equivalent to those given.

A natural topology on the space of left orderings of an arbitrary semi-group is introduced here. This space is proved to be compact, and for free abelian groups it is shown to be homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases.