The main purpose of this paper is to introduce a method to “stabilise” certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group G, namely Hom(ℤn , G). We show that this stabilised space of homomorphisms decomposes after suspending once with “summands” which can be reassembled, in a sense to be made precise below, into the individual spaces Hom(ℤn , G) after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilised space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilised space also allows the description of the additive reduced homology of the individual spaces Hom(ℤn , G), with the order of the Weyl group inverted.

We show that every continuous simple curve with σ-finite length has a tangent at positively many points. We also apply this result to functions with finite lower scaled oscillation; and study the validity of the results in higher dimension.

The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemerédi regularity lemma in graph theory. It shows that for any abelian group G and any bounded function f : G → [0, 1], there exists a subgroup H ⩽ G of bounded index such that, when restricted to most cosets of H, the function f is pseudorandom in the sense that all its nontrivial Fourier coefficients are small. Quantitatively, if one wishes to obtain that for 1 − ε fraction of the cosets, the nontrivial Fourier coefficients are bounded by ε, then Green shows that |G/H| is bounded by a tower of twos of height 1/ε3. He also gives an example showing that a tower of height Ω(log 1/ε) is necessary. Here, we give an improved example, showing that a tower of height Ω(1/ε) is necessary.

Gruenberg and Linnell showed that the standard relation module of a free product of n groups of the form Cr × $\mathbb{Z}$ could be generated by just n + 1 generators, raising the possibility of a relation gap. We explicitly give such a set of generators.

A limit of rational varieties need not be rational, even when all varieties in the family are projective and have at most terminal singularities.

Let E/Q be an elliptic curve, p a prime and K ∞/K the anticyclotomic Zp -extension of a quadratic imaginary field K satisfying the Heegner hypothesis. In this paper we give a new proof to a theorem of Bertolini which determines the value of the Λ-corank of Selp∞ (E/K ∞) in the case where E has ordinary reduction at p. In the case where E has supersingular reduction at p we make a conjecture about the structure of the module of Heegner points mod p. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines the value of the Λ-corank of Selp∞ (E/K ∞) in the case where E has supersingular reduction at p.

We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.

A theory of (co)homologies related to set-theoretic n-simplex relations is constructed in analogy with the known quandle and Yang–Baxter (co)homologies, with emphasis made on the tetrahedron case. In particular, this permits us to generalise Hietarinta's idea of “permutation-type” solutions to the quantum (or “tensor”) n-simplex equations. Explicit examples of solutions to the tetrahedron equation involving nontrivial cocycles are presented.

We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrised families of self-similar sets and measures in the complex plane, extending earlier results.

Let G be a connected perfect real Lie group. We show that there exists α < dim G and p ∈ $\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure on G, then the pth convolution power μ*p is absolutely continuous with respect to the Haar measure on G, with arbitrarily smooth density. As an application, we obtain that if A ⊂ G is a Borel set with Hausdorff dimension at least α, then the p-fold product set Ap contains a non-empty open set.

We consider 3-webs, hyper-para-complex structures and integrable Segre structures on manifolds of even dimension and generalise the second heavenly Plebański equation in the context of higher-dimensional hyper-para-complex structures. We also characterise the Segre structures admitting a compatible hyper-para-complex structure in terms of systems of ordinary differential equations.

We examine the common null spaces of families of Herz–Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in [1] can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.

For each diagram D of a 2-knot, we provide a way to construct a new diagram D′ of the same knot such that any sequence of Roseman moves between D and D′ necessarily involves branch points. The proof is done by developing the observation that no sphere eversion can be lifted to an isotopy in 4-space.

Among monoidal categories with finite coproducts preserved by tensoring on the left, we characterise those with finite biproducts as being precisely those in which the initial object and the coproduct of the unit with itself admit right duals. This generalises Houston's result that any compact closed category with finite coproducts admits biproducts.

We show that the proportion of polynomials of degree n over the finite field with q elements, which have a divisor of every degree below n, is given by c q n −1 + O(n −2). More generally, we give an asymptotic formula for the proportion of polynomials, whose set of degrees of divisors has no gaps of size greater than m. To that end, we first derive an improved estimate for the proportion of polynomials of degree n, all of whose non-constant divisors have degree greater than m. In the limit as q → ∞, these results coincide with corresponding estimates related to the cycle structure of permutations.

Let (M, ϕ) = (M 1, ϕ1) * (M 2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M 1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M 1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M 1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M 1 ⊂ M in arbitrary free product von Neumann algebras.

Let Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log1+ϵx] contain E 3 numbers, and almost all intervals [x,x + log3.51x] contain E 2 numbers. By this we mean that there are only o(X) integers 1 ⩽ x ⩽ X for which the mentioned intervals do not contain such numbers. The result for E 3 numbers is optimal up to the ϵ in the exponent. The theorem on E 2 numbers improves a result of Harman, which had the exponent 7 + ϵ in place of 3.51. We also consider general Ek numbers, and find them on intervals whose lengths approach log x as k → ∞.

We generalise the classical Bombieri–Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extension L/$\mathbb{Q}$ exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modular L-function L(s, f), the fundamental discriminants d for which the d-quadratic twist of L(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.

Let Γ be a nonuniform lattice acting on the real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of Γ in SO(n, 1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the representation variety such that the volume of a representation is constant on connected components of the semialgebraic subset. Combining our approach with the main result of [2] gives a new proof of the local rigidity theorem for nonuniform hyperbolic lattices and the analogue of Soma's theorem, which shows that the number of orientable hyperbolic manifolds dominated by a closed, connected, orientable 3-manifold is finite, for noncompact 3-manifolds.

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.