We provide an extension of the transference results of Beresnevich and Velani connecting homogeneous and inhomogeneous Diophantine approximation on manifolds and provide bounds for inhomogeneous Diophantine exponents of affine subspaces and their nondegenerate submanifolds.

We consider Carleson's problem regarding pointwise convergence for the Schrödinger equation. Bourgain proved that there is initial data, in Hs(ℝn) with $s<\frac{n}{2(n+1)}$, for which the solution diverges on a set of nonzero Lebesgue measure. We provide a different example enabling the generalisation to fractional Hausdorff measure.

The Morrison–Kawamata Cone Conjecture predicts that the action of the automorphism group on the effective nef cone and the action of the pseudo-automorphism group on the effective movable cone of a klt Calabi–Yau pair have rational, polyhedral fundamental domains. In [CPS], we proved the conjecture for certain blowups of Fano manifolds of index n - 1. In this paper, we consider the Morrison–Kawamata conjecture for blowups of Fano manifolds of index n - 2.

Inspired by a method of La Bretèche relying on some unique factorisation, we generalise work of Blomer, Brüdern and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the singular projective varieties defined by the following equation

$$ x_1 y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0 $$

Amicable pairs for a fixed elliptic curve defined over ℚ were first considered by Silverman and Stange where they conjectured an order of magnitude for the function that counts such amicable pairs. This was later refined by Jones to give a precise asymptotic constant. The author previously proved an upper bound for the average number of amicable pairs over the family of all elliptic curves. In this paper we improve this result to an asymptotic for the average number of amicable pairs for a family of elliptic curves.

In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we give a geometric proof of a Theorem due to Alvarez and Rodriguez that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half-planes attached along simple open geodesic components. We next consider gluing infinitely many pairs of pants along their cuffs to obtain an infinite hyperbolic surface. We prove that there always exists a choice of twists in the gluings such that the surface is complete regardless of the size of the cuffs. This generalises the examples of Matsuzaki.

In the second part we consider complete hyperbolic flute surfaces with rapidly increasing cuff lengths and prove that the corresponding quasiconformal Teichmüller space is incomplete in the length spectrum metric. Moreover, we describe the twist coordinates and convergence in terms of the twist coordinates on the closure of the quasiconformal Teichmüller space.

The polynomial Freĭman–Ruzsa conjecture over the integers is often phrased in terms of convex progressions. We give an alternative, apparently stronger formulation in terms of the more restrictive “ellipsoid progressions”, and show that these formulations are in fact equivalent. The key input to the equivalence proof comes from strong results in asymptotic convex geometry.

We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on the p-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the form y2 = f(x), under some simplifying hypotheses.

We show that for any constant Δ ≥ 2, there exists a graph Γ with O(nΔ / 2) vertices which contains every n-vertex graph with maximum degree Δ as an induced subgraph. For odd Δ this significantly improves the best-known earlier bound and is optimal up to a constant factor, as it is known that any such graph must have at least Ω(nΔ/2) vertices.

The hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalise this to the simplicial volume of link complements by analysing the corresponding toroidal decompositions. We then use it to prove a refined upper bound for the volume in terms of twists of various lengths for links.

Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z and w; let $\mathcal{X}$ be the generic element of the family of surfaces in ℙ given by

\begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*}

$\mathcal{X}$

$\mathcal{X}$

$\mathcal{X}$

The magnitude of a graph is one of a family of cardinality-like invariants extending across mathematics; it is a cousin to Euler characteristic and geometric measure. Among its cardinality-like properties are multiplicativity with respect to cartesian product and an inclusion-exclusion formula for the magnitude of a union. Formally, the magnitude of a graph is both a rational function over ℚ and a power series over ℤ. It shares features with one of the most important of all graph invariants, the Tutte polynomial; for instance, magnitude is invariant under Whitney twists when the points of identification are adjacent. Nevertheless, the magnitude of a graph is not determined by its Tutte polynomial, nor even by its cycle matroid, and it therefore carries information that they do not.

Denote ΘC as the Frobenius class of a curve C over the finite field 𝔽q. In this paper we determine the expected value of Tr(ΘCn) where C runs over all biquadratic curves when q is fixed and g tends to infinity. This extends work done by Rudnick [15] and Chinis [5] who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda [1] who looked at ℓ-cyclic curves, for ℓ a prime, as well as cubic non-Galois curves.

Motivated by the results of Scott and Patel about “untangling” closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou–Rabee connects the primitivity index function fprim(n, FN) to the residual finiteness growth function for FN.

We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target property. This generalises several similar results for β-transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.

Let be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

There is a well-known analogy between integers and polynomials over 𝔽q, and a vast literature on analytic number theory for polynomials. From a geometric point of view, polynomials are equivalent to effective 0-cycles on the affine line. This leads one to ask: Can the analogy between integers and polynomials be extended to 0-cycles on more general varieties? In this paper we study prime factorisation of effective 0-cycles on an arbitrary connected variety V over 𝔽q, emphasizing the analogy between integers and 0-cycles. For example, inspired by the works of Granville and Rhoades, we prove that the prime factors of 0-cycles are typically Poisson distributed.

We prove a robust version of Freiman's 3k – 4 theorem on the restricted sumset A+ΓB, which applies when the doubling constant is at most (3+$\sqrt{5}$)/2 in general and at most 3 in the special case when A = −B. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz–Sobolev inequality.

In this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1 × ℙ1 and, more recently, in (ℙ1)r. In ℙ1 × ℙ1 the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm × ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1 × ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting.

In this paper we introduce and investigate the notions of diagrams and discrete extensions in the study of finitary 2-representations of finitary 2-categories.