AbstractWe correct mistakes in Lemma 8 and in the proof of Lemma 10 from (Bull. Lond. Math. Soc. 49 (2017), no. 6, 965–978).

AbstractLet be a holomorphic quadratic differential of finite norm in the unit disk . Let be equipped with the metric . In this paper, we prove that a geodesic‐preserving map of the unit disk into itself is affine. A map is called geodesic‐preserving if the image of each geodesic is a geodesic. We make no assumptions on the continuity of such a map.

AbstractThe parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct them, and is responsible for a wide range of unexplained arithmetic phenomena. It is one of the main consequences of the Birch and Swinnerton‐Dyer conjecture and lets one calculate the parity of the rank of an elliptic curve using root numbers. In this handbook, we explain how to use local root numbers of elliptic curves to realise some of these phenomena, with an emphasis on explicit calculations. The text is aimed at a ‘user’ and, as such, we will not be concerned with the proofs of known cases of the parity conjecture, but instead, we will demonstrate the use of the theory by means of examples.

AbstractWe show that the ring of integers of is existentially definable in the ring of integers of , where denotes the field of all totally real numbers. This implies that the ring of integers of is undecidable and first‐order nondefinable in . More generally, when is a totally imaginary quadratic extension of a totally real field , we use the unit groups of orders to produce existentially definable totally real subsets . Under certain conditions on , including the so‐called ‐number of being the minimal value , we deduce the undecidability of . This extends previous work that proved an analogous result in the opposite case . In particular, unlike prior work, we do not require that contains only finitely many roots of unity.

AbstractWe show that a free‐by‐cyclic group with a polynomially growing monodromy is subgroup separable exactly when it is virtually . We also prove that random deficiency 1 groups are not subgroup separable with positive asymptotic probability.

AbstractLet be an imaginary quadratic field and its ring of integers. Fix a rational integer . A set is called a ‐‐tuple in if , where for all such that . Here, we prove the non‐existence of ‐‐tuples in for and .

AbstractFor any complex K3 surface , we construct a one‐dimensional deformation in which all integers with occur as Picard numbers of some fibres. In contrast, we prove that the generic one‐dimensional local family of K3 surfaces admits only 0 and 1 as Picard numbers of the fibres.

AbstractWe show that the Gaussian kernel on any nonsimply connected closed Riemannian manifold , where is the geodesic distance, is not positive definite for any , combining analyses in the recent preprint  by Da Costa–Mostajeran–Ortega and classical comparison theorems in Riemannian geometry.

AbstractLet be an even Borel probability measure on . For every , consider independent random vectors in , with independent coordinates having distribution . We establish a sharp threshold for the product measure of the random polytope in under the assumption that the Legendre transform of the logarithmic moment generating function of satisfies the condition where . An application is a sharp threshold for the case of the product measure , with density , where is the ‐norm and .

AbstractWe show that Connes' metric on the state space associated with a spectral triple is nowhere infinite exactly when it is globally bounded. Moreover, we produce a family of simple examples showing that this is not automatically the case.