We show that the property of a C*-algebra that all its Hilbert modules have a frame, in the case of σ-unital C*-algebras, is preserved under Rieffel–Morita equivalence. In particular, we show that a σ-unital continuous-trace C*-algebra with trivial Dixmier–Douady class, all of whose Hilbert modules admit a frame, has discrete spectrum. We also show this for the tensor product of any commutative C*-algebra with the C*-algebra of compact operators on any Hilbert space.

We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic subgroup of GL2 (defined by F. Momose). We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.

We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite-dimensional manifold. We succeed for a particular type of piecewise-smooth loops. We also examine the action of the diffeomorphism group of the circle. It is not a useful action on the manifold that we define. We consider how one might fix this problem and conclude that it can only be done by completing to the space of loops of bounded variation.

We augment the body of existing results on embedding finite semigroups of a certain type into 2-generator finite semigroups of the same type. The approach adopted applies to finite semigroups the idempotents of which form a band and also to finite orthodox semigroups.

We give an explicit description of the stable reduction of superelliptic curves of the form yn=f(x) at primes $\mathfrak{p}$ whose residue characteristic is prime to the exponent n. We then use this description to compute the local L-factor and the exponent of conductor at $\mathfrak{p}$ of the curve.

We prove that the classical universal Taylor series in the complex plane are never frequently universal. On the other hand, we prove the 1-upper frequent universality of all these universal Taylor series.

In this paper, we study Abelian and metabelian quotients of braid groups of oriented surfaces with boundary components. We provide group presentations and we prove rigidity results for these quotients arising from exact sequences related to (generalised) Fadell–Neuwirth fibrations.

We present a ready to compute trace formula for Hecke operators on vector-valued modular forms of integral weight for SL2(ℤ) transforming under the Weil representation. As a corollary, we obtain a ready to compute dimension formula for the corresponding space of vector-valued cusp forms, which is more general than the dimension formulae previously published in the vector-valued setting.

We suggest a way to quantize, using Berezin–Toeplitz quantization, a compact hyperkähler manifold (equipped with a natural 3-plectic form), or a compact integral Kähler manifold of complex dimension n regarded as a (2n−1)-plectic manifold. We show that quantization has reasonable semiclassical properties.

In a previous paper, we studied the homogenized enveloping algebra of the Lie algebra sℓ(2,ℂ) and the homogenized Verma modules. The aim of this paper is to study the homogenization $\mathcal{O}$B of the Bernstein–Gelfand–Gelfand category $\mathcal{O}$ of sℓ(2,ℂ), and to apply the ideas developed jointly with J. Mondragón in our work on Groebner basis algebras, to give the relations between the categories $\mathcal{O}$B and $\mathcal{O}$ as well as, between the derived categories $\mathcal{D}$b($\mathcal{O}$B) and $\mathcal{D}$b($\mathcal{O}$).

This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space X which are obtained by deleting singletons determine X uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible. We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals ℝ, the rationals ℚ and the irrationals ℙ are reconstructible, as well as spaces occurring as Stone–Čech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set C.

In this paper, we show that the known models for (∞, 1)-categories can all be extended to equivariant versions for any discrete group G. We show that in two of the models we can also consider actions of any simplicial group G.

In this note, we give a new simple construction of all maximal abelian ideals in a Borel subalgebra of a complex simple Lie algebra. We also derive formulas for dimensions of certain maximal abelian ideals in terms of the theory of Borel de Siebenthal.

Let f,g:(ℝn, 0) → (ℝ, 0) be Cr+1 functions, r ∈ ℕ. We will show that if ∇f(0)=0 and there exist a neighbourhood U of 0 ∈ ℝn and a constant C > 0 such that

$$\begin{equation*} \left|\partial^m(g-f)(x)\right| ≤ C \left|\nabla f(x)\right|^{r+2-|m|} \quad \textrm{ for } x\in U, \end{equation*} $$