Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in A if they are the vertices of a convex polytope, which contains no points of A in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t,m,s)-nets or (t,s)-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of (t,m,s)-nets, obeying a relaxed equidistribution condition.

We define and study the space of q-opers associated with Bethe equations for integrable models of XXZ type with quantum toroidal algebra symmetry. Our construction is suggested by the study of the enumerative geometry of cyclic quiver varieties, in particular the ADHM moduli spaces. We define $\left (\overline {GL}(\infty ),q\right )$-opers with regular singularities and then, by imposing various analytic conditions on singularities, arrive at the desired Bethe equations for toroidal q-opers.

A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$. The previous best bound was $O(\Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$. The best previous bound for this result was exponential in $\Delta$.

In the first part of the paper, we use states on $C^{*}$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^{*}$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^{*}$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^{*}$-module under the assumption that their inner product has a negative real part. We introduce the concept of Pythagoras orthogonality and discuss its properties. We describe this notion for Hilbert space operators in terms of the parallelogram law and some limit conditions. We present several examples in order to illustrate the relationship between the Birkhoff–James, Roberts, and Pythagoras orthogonalities, and the usual orthogonality in the framework of Hilbert $C^{*}$-modules.

Since Faltings proved Mordell’s conjecture in [16] in 1983, we have known that the sets of rational points on curves of genus at least $2$ are finite. Determining these sets in individual cases is still an unsolved problem. Chabauty’s method (1941) [10] is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the Jacobian, the closure of the Mordell–Weil group with the p-adic points of the curve. Under the condition that the Mordell–Weil rank is less than the genus, Chabauty’s method, in combination with other methods such as the Mordell–Weil sieve, has been applied successfully to determine all rational points in many cases.

Minhyong Kim’s nonabelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Besser, Dogra, Müller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both $3$).

This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only ‘simple algebraic geometry’ (line bundles over the Jacobian and models over the integers).