Let Σg be a compact, connected, orientable surface of genus g ≥ 2. We ask for a parametrisation of the discrete, faithful, totally loxodromic representations in the deformation space Hom(π1(Σg), SU(3, 1))/SU(3, 1). We show that such a representation, under some hypothesis, can be determined by 30g − 30 real parameters.

We establish that any subset of ℝd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d ⩾ 4.

We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δk1 and Δk2 are two fixed non-degenerate simplices of k1 + 1 and k2 + 1 points respectively, then any subset of ℝd of positive upper Banach density with d ⩾ k1 + k2 + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Δk1 × Δk2.

A new direct proof of the fact that any subset of ℝd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k + 1 points provided d ⩾ k + 1, a result originally due to Bourgain, is also presented.

The initial value problems for some semilinear wave and heat equations are investigated in two space dimensions. By proving the existence of ground state, strong instability of standing waves for the associated wave and heat equations are obtained.

We give general estimates for the approximation numbers of composition operators on the Hardy space on the ball Bd and the polydisk ${\mathbb D}$d and of composition operators on the Bergman space on the polydisk.

We study entire functions whose zeros and one-points lie on distinct finite systems of rays. General restrictions on these rays are obtained. Non-trivial examples of entire functions with zeros and one-points on different rays are constructed, using the Stokes phenomenon for second order linear differential equations.

Let K be an algebraically closed field and $\mathbb{A}$n ≅ Kn affine n-space. It is known that a finite group $\frak{G}$ can only act freely on $\mathbb{A}$n if K has characteristic p > 0 and $\frak{G}$ is a p-group. In that case the group action is “non-linear” and the ring of regular functions K[$\mathbb{A}$n] must be a trace-surjectiveK − $\frak{G}$-algebra.

Now let k be an arbitrary field of characteristic p > 0 and let G be a finite p-group. In this paper we study the category $\mathfrak{Ts}$ of all finitely generated trace-surjective k − G algebras. It has been shown in [13] that the objects in $\mathfrak{Ts}$ are precisely those finitely generated k − G algebras A such that AG ≤ A is a Galois-extension in the sense of [7], with faithful action of G on A. Although $\mathfrak{Ts}$ is not an abelian category it has “s-projective objects”, which are analogues of projective modules, and it has (s-projective) categorical generators, which we will describe explicitly. We will show that s-projective objects and their rings of invariants are retracts of polynomial rings and therefore regular UFDs. The category $\mathfrak{Ts}$ also has “weakly initial objects”, which are closely related to the essential dimension of G over k. Our results yield a geometric structure theorem for free actions of finite p-groups on affine k-varieties. There are also close connections to open questions on retracts of polynomial rings, to embedding problems in standard modular Galois-theory of p-groups and, potentially, to a new constructive approach to homogeneous invariant theory.

We compute certain twists of the classical modular curve X(8). Searching for rational points on these twists enables us to find non-trivial pairs non-isogenous elliptic curves over ℚ whose 8-torsion subgroups are isomorphic as Galois modules. We also show that there are infinitely many examples over ℚ.

We introduce an embedding of the Torelli group of a compact connected oriented surface with non-empty connected boundary into the completed Kauffman bracket skein algebra of the surface, which gives a new construction of the first Johnson homomorphism.

We show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, SU(p, q), Sp(2n + 2, $\mathbb{R}$), SO*(2n + 2), SO(2n, 2). This reproves or generalises some results in [2, 9, 11, 15].