We analyze the computational complexity of an algorithm to solve the conjugacy search problem in a certain family of metabelian groups. We prove that in general the time complexity of the conjugacy search problem for these groups is at most exponential. For a subfamily of groups, we prove that the conjugacy search problem is polynomial. We also show that for a different subfamily the conjugacy search problem reduces to the discrete logarithm problem.

It is not known whether the inverse of a frequently hypercyclic bilateral weighted shift on c0(ℤ) is again frequently hypercyclic. We show that the corresponding problem for upper frequent hypercyclicity has a positive answer. We characterise, more generally, when bilateral weighted shifts on Banach sequence spaces are (upper) frequently hypercyclic.

Functorial semi-norms on singular homology give refined ‘size’ information on singular homology classes. A fundamental example is the ℓ1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the ℓ1-semi-norm.

We study Higgs bundles over an elliptic curve with complex reductive structure group, describing the (normalisation of) its moduli spaces and the associated Hitchin fibration. The case of trivial degree is covered by the work of Thaddeus in 2001. Our arguments are different from those of Thaddeus and cover arbitrary degree.

We calculate the endomorphism dga of Franke's exotic algebraic model for the K-local stable homotopy category at odd primes. We unravel its original abstract structure to give explicit generators, differentials and products.

Let G be a simple, simply connected, compact Lie group, and let M be an orientable, smooth, connected, closed 4-manifold. In this paper, we calculate the homotopy type of the suspension of M and the homotopy types of the gauge groups of principal G-bundles over M when π1(M) is (1) ℤ*m, (2) ℤ/prℤ, or (3) ℤ*m*(*nj=1ℤ/prjjℤ), where p and the pj's are odd primes.

Let K be a global field, $\mathcal{V}$ a proper subset of the set of all primes of K, $\mathcal{S}$ a finite subset of $\mathcal{V}$, and ${\tilde K}$ (resp. Ksep) a fixed algebraic (resp. separable algebraic) closure of K with $K_\mathrm{sep}\{\subseteq}{\tilde K}$. Let Gal(K) = Gal(Ksep/K) be the absolute Galois group of K. For each $\mathfrak{p}\in\mathcal{V}$, we choose a Henselian (respectively, a real or algebraic) closure $K_\mathfrak{p}$ of K at $\mathfrak{p}$ in ${\tilde K}$ if $\mathfrak{p}$ is non-archimedean (respectively, archimedean). Then, $K_{\mathrm{tot},\mathcal{S}}=\bigcap_{\mathfrak{p}\in\mathcal{S}}\bigcap_{\tau\in{\rm Gal}(K)}K_\mathfrak{p}^\tau$ is the maximal Galois extension of K in Ksep in which each $\mathfrak{p}\in\mathcal{S}$ totally splits. For each $\mathfrak{p}\in\mathcal{V}$, we choose a $\mathfrak{p}$-adic absolute value $|~|_\mathfrak{p}$ of $K_\mathfrak{p}$ and extend it in the unique possible way to ${\tilde K}$. Finally, we denote the compositum of all symmetric extensions of K by Ksymm. We consider an affine absolutely integral variety V in $\mathbb{A}_K^n$. Suppose that for each $\mathfrak{p}\in\mathcal{S}$ there exists a simple $K_\mathfrak{p}$-rational point $\mathbf{z}_\mathfrak{p}$ of V and for each $\mathfrak{p}\in\mathcal{V}\smallsetminus\mathcal{S}$ there exists $\mathbf{z}_\mathfrak{p}\in V({\tilde K})$ such that in both cases $|\mathbf{z}_\mathfrak{p}|_\mathfrak{p}\le1$ if $\mathfrak{p}$ is non-archimedean and $|\mathbf{z}_\mathfrak{p}|_\mathfrak{p}<1$ if $\mathfrak{p}$ is archimedean. Then, there exists $\mathbf{z}\in V(K_{\mathrm{tot},\mathcal{S}}\cap K_\mathrm{symm})$ such that for all $\mathfrak{p}\in\mathcal{V}$ and for all τ ∈ Gal(K), we have $|\mathbf{z}^\tau|_\mathfrak{p}\le1$ if $\mathfrak{p}$ is archimedean and $|\mathbf{z}^\tau|_\mathfrak{p}<1$ if $\mathfrak{p}$ is non-archimedean. For $\mathcal{S}=\emptyset$, we get as a corollary that the ring of integers of Ksymm is Hilbertian and Bezout.

The genus spectrum of a finite group G is the set of all g such that G acts faithfully on a compact Riemann surface of genus g. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the Abelian p-groups. Motivated by earlier work of Talu for odd primes, we develop a general combinatorial method, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of Abelian p-groups, including the reduced minimum genus. In particular, we determine the complete genus spectrum for a large subclass, namely, those having ‘large’ defining invariants. With our method we construct infinitely many counterexamples to a conjecture of Talu, which states that an Abelian p-group is recoverable from its genus spectrum. Finally, we give a series of examples of our method, in the course of which we prove, for example, that almost all elementary Abelian p-groups are uniquely determined by their minimum genus, and that almost all Abelian p-groups of exponent p2 are uniquely determined by their minimum genus and Kulkarni invariant.

We characterise singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.

In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation

$$\begin{equation*} \left\{ \begin{array}{l} \epsilon^{2\alpha}(-\Delta)_\rho^{\alpha} u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\ u\in H^{\alpha}(\mathbb{R}^n) \end{array} \right. \end{equation*}$$

$1<p<\frac{n+2\alpha}{n-2\alpha}$

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E, and let M be a τC-rigid module with U its Bongartz τC-complement. If the induced module, M ⊗CB, is τB-rigid, we give a necessary and sufficient condition for U ⊗CB to be its Bongartz τB-complement. If M is τB-rigid, we again provide a necessary and sufficient condition for U ⊗CB to be its Bongartz τB-complement.

We prove a Freed–Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa–Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the PU(2)-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms to the Vafa–Witten equations, and prove that the moduli space of solutions to the perturbed Vafa–Witten equations on a closed symplectic four-manifold for the structure group SU(2) or SO(3) is a smooth manifold of dimension zero for a generic choice of the perturbation parameters.

A norm ‖ċ‖ on c00 is called combinatorial if there is a regular family of finite subsets $\mathcal{F}$, so that $\|x\|=\sup_{F \in \mathcal{F}} \sum_{i \in F} |x(i)|$. We prove the set of extreme points of the ball of a combinatorial Banach space is countable. This extends a theorem of Shura and Trautman. The second contribution of this article is to exhibit many new examples of extreme points for the unit ball of dual Tsirelson's original space and give an explicit construction of an uncountable collection of extreme points of the ball of Tsirelson's original space. We also prove some stability properties of the intermediate norms used to define Tsirelson's space and give a lower bound of the stabilization function for these intermediate norms.