We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley.

Let A ⊆ [1, N] be a set of integers with |A| ≫ $\sqrt N$ . We show that if A avoids about p/2 residue classes modulo p for each prime p, then A must correlate additively with the squares S = {n2 : 1 ≤ n ≤ $\sqrt N$ }, in the sense that we have the additive energy estimate $$ E(A,S)\gg N\log N. $$ This is, in a sense, optimal.

Let G be a permutation group on a set Ω. A subset of Ω is a base for G if its pointwise stabiliser in G is trivial. In this paper we introduce and study an associated graph Σ(G), which we call the Saxl graph of G. The vertices of Σ(G) are the points of Ω, and two vertices are adjacent if they form a base for G. This graph encodes some interesting properties of the permutation group. We investigate

We run an iteration argument due to Pramanik and Seeger, to provide a proof of sharp decoupling inequalities for conical surfaces and for k-cones. These are extensions of results of Łaba and Pramanik to sharp exponents.

In this paper, we study orbifold constructions associated with the Leech lattice vertex operator algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge 24 is uniquely determined by its weight one Lie algebra if the Lie algebra has the type A3,43A1,2, A4,52, D4,12A2,6, A6,7, A7,4A1,13, D5,8A1,2 or D6,5A1,12 by using the reverse

This classification is found by analyzing the action of a normal subgroup of B3 as hyperbolic isometries. This paper gives an example of an unfaithful specialisation of the Burau representation on B4 that is faithful when restricted to B3, as well as examples of unfaithful specialisations of B3.

We prove that every flat tensor-idempotent in the module category Mod- of a tensor-triangulated category comes from a unique smashing ideal in . We deduce that the lattice of smashing ideals forms a frame.

Triple homomorphisms on C*-algebras and JB*-triples have been studied in the literature. From the viewpoint of associative algebras, we characterise the structure of triple homomorphisms from an arbitrary ⋆-algebra onto a prime *-algebra. As an application, we prove that every triple homomorphism from a Banach ⋆-algebra onto a prime semisimple idempotent Banach *-algebra is continuous. The analogous

It is well known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group G can be detected on the cohomology group H1(G,R[G]), where R is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group G the same information

It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k-stable. Then there is a normal subgroup H ≤ G of index at most n, and a set Y ⊆ G, which is a union of cosets of H, such that |A △ Y| ≤ε|H|. It follows that, for any coset C of H, either |C ∩ A|≤ ε|H| or |C \ A| ≤ ε |H|. This qualitatively generalises recent work of

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We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field K of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction

Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander–Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points.

For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the

We show that a monotone Lagrangian L in ${\mathbb{C}}{\mathbb{P}}^n$ of minimal Maslov number n + 1 is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to ${\mathbb{R}}{\mathbb{P}}^n$ . To prove this we use Zapolsky’s canonical pearl complex for L over ${\mathbb{Z}}$ , and twisted versions thereof, where the twisting is determined by connected covers of L. The main tool is

We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function

Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate

We construct two infinite series of Moufang loops of exponent 3 whose commutative centre (i. e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of orders 38 and 311 one of which can be defined as the Moufang triplication of the free Burnside group B(3, 3).

The Fatou–Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares

Let P be a finitely generated ideal of a commutative ring R. Krull's principal ideal theorem states that if R is Noetherian and P is minimal over a principal ideal of R, then P has height at most one. Straightforward examples show that this assertion fails if R is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem.

We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

We describe a construction which associates to any median metric space a pseudometric satisfying the binary intersection property for closed balls. Under certain conditions, this implies that the resulting space is, in fact, an injective metric space, bilipschitz equivalent to the original metric. In the course of doing this, we derive a few other facts about median metrics, and the geometry of CAT(0)

Given any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.

Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$ [G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of

Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior (A + Γ)°, when Γ is a piecewise ${\mathcal C}^2$ curve and A ⊂ ℝ2. To begin, we give an example of a very large (full-measure, dense, Gδ) set A such that (A + S1)° = ∅, where S1 denotes the unit circle. This suggests that merely the size of A does not guarantee

In recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting

We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric

In this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of quadratic twists of all optimal elliptic curves over ${\mathbb Q}$ .

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In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length n > 4 is commensurable to

We investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽q[t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free

We study the iteration of transcendental self-maps of $\,\mathbb{C}^*\!:=\mathbb{C}\setminus \{0\}$ , that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in

Following ideas that go back to Cannon, we show the rationality of various generating functions of growth sequences counting embeddings of convex subgraphs in locally-finite, vertex-transitive graphs with the (relative) falsification by fellow traveler property (fftp). In particular, we recover results of Cannon, of Epstein, Iano–Fletcher and Zwick, and of Calegari and Fujiwara. One of our applications

Let $\mathcal{M}$ denote the mapping class group of Σ, a compact connected oriented surface with one boundary component. The action of $\mathcal{M}$ on the nilpotent quotients of π1(Σ) allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of $\mathcal{M}$ , called the Lagrangian filtration. He also introduced a version of the Johnson

We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates

A new homological dimension, called rigidity dimension, is introduced to measure the quality of resolutions of finite dimensional algebras (especially of infinite global dimension) by algebras of finite global dimension and big dominant dimension. Upper bounds of the dimension are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological

In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the

The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝd, such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every graph with maximum degree Δ has separation dimension less than 20Δ, which is best possible up to a constant

Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$ . Given a modular Galois representation

We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups

We consider fibrewise singly generated Fell bundles over étale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo–Martin–Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the

We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: we construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.

We give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula

We study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G→ Γ that make (G, Γ) into matched pair of groups endowed with a natural crossed action on 𝒟 such that 𝒞 is equivalent to a certain associated crossed extension 𝒟(G

The first purpose of our paper is to show how Hooley’s celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy–Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series.

We show that provided that n is a power of two, any nD measures in ℝn can be bisected by an arrangement of D hyperplanes.

Paraconformal or GL(2, ℝ) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n – 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of GL(2, ℝ) structures, namely dispersionless integrable

We show that if A is a finite K-approximate subgroup of an s-step nilpotent group then there is a finite normal subgroup $H \subset {A^{{K^{{O_s}(1)}}}}$ modulo which ${A^{{O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)}}$ contains a nilprogression of rank at most ${O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)$ and size at least $\exp ( - {O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K))|A|$ . This partially

We construct links of arbitrarily many components each component of which is slice and yet are not concordant to any link with even one unknotted component. The only tool we use comes from the Alexander modules.

Over the past forty years many papers have studied logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. Terao conjectured thirty years ago that when a curve is a finite set of distinct lines (i

We show that for all large enough x the interval [x, x + x1/2 log1.39x] contains numbers with a prime factor p > x18/19. Our work builds on the previous works of Heath–Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals [x, x + x1/2 + ϵ]. We also incorporate some ideas from Harman’s book Prime-detecting sieves (2007). The main new ingredient that we use

Let ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition

Given a positive integer n let ω (n) denote the number of distinct prime factors of n, and let a be fixed positive integer. Extending work of Kubilius, we develop a bivariate probabilistic model to study the joint distribution of the deterministic vectors (ω(n), ω(n + a)), with n ≤ x as x → ∞, where n and n + a belong to a subset of ℕ with suitable properties. We thus establish a quantitative version

The residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable

We study C1-robustly transitive and nonhyperbolic diffeomorphisms having a partially hyperbolic splitting with one-dimensional central bundle whose strong un-/stable foliations are both minimal. In dimension 3, an important class of examples of such systems is given by those with a simple closed periodic curve tangent to the central bundle. We prove that there is a C1-open and dense subset of such