Bulletin of The London Mathematical Society

By analogy with the Cayley graph of a group with respect to a finite generating set or the Cayley–Abels graph of a totally disconnected, locally compact group, we detail countable connected graphs associated to Polish groups that we term Cayley–Abels–Rosendal graphs. A group admitting a Cayley–Abels–Rosendal graph acts on it continuously, coarsely metrically properly and cocompactly by isometries of the path metric. By an expansion of the Milnor–Schwarz lemma, it follows that the group is generated by a coarsely bounded set and the group equipped with a word metric with respect to a coarsely bounded generating set and the graph are quasi‐isometric. In other words, groups admitting Cayley–Abels–Rosendal graphs are topological analogues of finitely generated groups. Our goal is to introduce this topological perspective on the work of Rosendal to a geometric group theorist. We apply these concepts to homeomorphism groups of countable Stone spaces. We completely characterize when these homeomorphism groups are coarsely bounded, when they are locally bounded (all of them are), and when they admit a Cayley–Abels–Rosendal graph, and if so produce a coarsely bounded generating set.

Given a pair , consisting of a closed Riemann surface and a holomorphic connection on the trivial principal bundle , the Riemann–Hilbert map sends to its monodromy representation. We compute the derivative of this map, and provide a simple description of the locus where it is injective, recovering in the process several previously obtained results.

We prove global Calderón–Zygmund type estimates for parabolic quasiminimizers of integral functionals with ‐growth.

In this short note, we refine a result of Schiffmann–Vasserot, by showing that the localized preprojective cohomological Hall algebra of any quiver is spherical, that is, generated by elements of minimal dimension.

We obtain monotonicity and convexity results for the heat content of domains in Riemannian manifolds and in Euclidean space subject to various initial temperature conditions. We introduce the notion of a strictly decreasing temperature set , and show that it is a sufficient condition to ensure monotone heat content. In addition, in Euclidean space, we construct a domain and an initial condition for which the heat content is not monotone, as well as a domain and an initial condition for which the heat content is monotone but not convex.

In this note, we extend the boundary criterion for relative cubulation of the first author and Groves to the case when the peripheral subgroups are not necessarily one‐ended. Specifically, if the boundary criterion is satisfied for a relatively hyperbolic group, we show that, up to taking a refined peripheral structure , the group admits a relatively geometric action on a CAT(0) cube complex. We anticipate that this refinement will be useful for constructing new relative cubulations in a variety of settings.

Let be a hyperplane arrangement in a complex projective space. It is an open question if the degree one cohomology jump loci (with complex coefficients) are determined by the combinatorics of . By the work of Falk and Yuzvinsky [Compositio Math. 143 (2007), no. 4, 1069–1088] and Marco‐Buzunáriz [Graphs Combin. 25 (2009), no. 4, 469–488], all the irreducible components passing through the origin are determined by the multinet structure, which is combinatorially determined. Denham and Suciu introduced the pointed multinet structure, which is combinatorially determined, to obtain examples of arrangements with translated positive‐dimensional components in the degree one cohomology jump loci [Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1435–1470]. Suciu asked the question if all translated positive‐dimensional components appear in this manner [Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), no. 2, 417–481]. In this paper, we show that the double star arrangement introduced by Ishibashi, Sugawara, and Yoshinaga [Adv. in Appl. Math. 162 (2025), Paper No. 102790, Example 3.2] gives a negative answer to this question.

We study the behavior of the Hilbert–Kunz multiplicity of powers of an ideal in a local ring. In dimension 2, we provide answers to some problems raised by Smirnov, and give a criterion to answer one of his questions in terms of a “Ratliff–Rush version” of the Hilbert–Kunz multiplicity.

We investigate arcs on a pair of pants and present an algorithm to compute the self‐intersection number of an arc. Additionally, we establish bounds for the self‐intersection number in terms of the word length. We also prove that the spectrum of self‐intersection numbers of 2‐low‐lying arcs covers all natural numbers.

We devise an explicit method for computing combinatorial formulae for Hadamard products of certain rational generating functions. The latter arise naturally when studying so‐called ask zeta functions of direct sums of modules of matrices or class‐ and orbit‐counting zeta functions of direct products of nilpotent groups. Our method relies on shuffle compatibility of coloured permutation statistics and coloured quasisymmetric functions, extending recent work of Gessel and Zhuang.

Let be a finite‐dimensional algebra. In this paper, we show that there is a natural bijection between cosilting modules in and semibricks in satisfying some condition. Also this bijection restricts to a bijection between all semibricks in and a certain subclass of cosilting modules. These bijections are generalizations of Asai's correspondence ( Int. Math. Res. Not . 16 (2020) 4993–5054) between support ‐tilting modules and right finite semibricks.

Every noncompact surface is shown to have a (3,6)‐tight triangulation, and applications are given to the generic rigidity of countable bar‐joint frameworks in . In particular, every noncompact surface has a (3,6)‐tight triangulation that is minimally 3‐rigid. A simplification of Richards' proof of Kerékjártó's classification of noncompact surfaces is also given.

The class of strictly sign regular (SSR) matrices has been extensively studied by many authors over the past century, notably by Schoenberg, Motzkin, Gantmacher, and Krein. A classical result of Gantmacher–Krein assures the existence of SSR matrices for any dimension and sign pattern. In this article, we provide an algorithm to explicitly construct an SSR matrix of any given size and sign pattern. (We also provide in the Appendix, a Python code implementing our algorithm.) To develop this algorithm, we show that one can extend an SSR matrix by adding an extra row (column) to its border, resulting in a higher order SSR matrix. Furthermore, we show how inserting a suitable new row/column between any two successive rows/columns of an SSR matrix results in a matrix that remains SSR. We also establish analogous results for SSR matrices of order for any .

In 2005, Balmer defined the ringed space for a given tensor triangulated category, while in 2023, the second author introduced the ringed space for a given triangulated category. In the algebro‐geometric context, these spectra provided several reconstruction theorems using derived categories. In this paper, we prove that is an open ringed subspace of for a quasi‐projective variety . As an application, we provide a new proof of the Bondal–Orlov and Ballard reconstruction theorems in terms of these spectra. Recently, the first author introduced the Fourier–Mukai locus for a smooth projective variety , which is constructed by gluing Fourier–Mukai partners of inside . As another application of our main theorem, we demonstrate that can be viewed as an open ringed subspace of . As a result, we show that all the Fourier–Mukai partners of an abelian variety can be reconstructed by topologically identifying the Fourier–Mukai locus within .

Let be a regular ‐finite ring of prime characteristic . We prove that the injective dimension of every unit Frobenius module in the category of unit Frobenius modules is at most . We further show that for unit Cartier modules the same bound holds over any noetherian ‐finite ring of prime characteristic . This shows that is a uniform upper bound for the injective dimension of any unit Cartier module over a noetherian ‐finite ring .

If is a saturated fusion system on a finite ‐group , we define the Chern subring of to be the subring of generated by Chern classes of ‐stable representations of . We show that is contained in and apply a result of Green and the first author to describe its maximal ideal spectrum in terms of a certain category of elementary abelian subgroups. We obtain similar results for various related subrings, including those generated by characteristic classes of ‐stable ‐sets.

In this note, we show how canonical transformations reveal hidden convexity properties for deterministic optimal control problems, which in turn result in global existence of solutions to first‐order Hamilton–Jacobi–Bellman equations.

Geometric invariant theory (GIT) produces quotients of algebraic varieties by reductive groups. If the variety is projective, this quotient depends on a choice of polarisation; by work of Dolgachev–Hu and Thaddeus, it is known that two quotients of the same variety using different polarisations are related by birational transformations. Only finitely many birational varieties arise in this way: variation of GIT fails to capture the entirety of the birational geometry of GIT quotients. We construct a space parametrising all possible GIT quotients of all birational models of the variety in a simple and natural way, which captures the entirety of the birational geometry of GIT quotients in a precise sense. It yields in particular a compactification of a birational analogue of the set of stable orbits of the variety.

We derive an optimal power‐weighted Hardy‐type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the latter inequality differs from the former. Moreover, by iterating these inequalities we derive the sequence of power‐weighted Birman–Hardy–Rellich‐type inequalities in integral form on finite intervals and then also prove the analogous sequence of inequalities in differential form. We use the one‐dimensional Hardy‐type result in differential form to derive an optimal multi‐dimensional version of the power‐weighted Hardy inequality in differential form on annuli (i.e., spherical shell domains), and once more employ an iteration procedure to derive the Birman–Hardy–Rellich‐type sequence of power‐weighted higher order Hardy‐type inequalities for annuli. In the limit as the annulus approaches \{0}, we recover well‐known prior results on Rellich‐type inequalities on \{0}.

We study an integro‐differential free transmission problem associated with the Bellman–Isaacs‐type operator that is solution‐dependent. The existence of a viscosity solution is proved by constructing solutions of suitable auxiliary problems for such a nonlocal problem. We also identify circumstances under which the gradient of the solution enjoys an interior Hölder regularity whose estimates remain uniform as the degree of the equation approaches 2.

In recent work of Chan–Huang–Lee, it is shown that if a manifold enjoys uniform bounds on (a) the negative part of the scalar curvature, (b) the local entropy, and (c) volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially (depending only on these a priori bounds). In this work, we show that the bound on scalar curvature assumption (a) is redundant. We also give some applications of this quantitative short‐time existence, including a Ricci flow smoothing result for measure space limits, a Gromov–Hausdorff compactness result, and a topological and geometric rigidity result in the case that the a priori local bounds are strengthened to be global.

This work demonstrates classical generation is preserved by the derived pushforward along the structure morphism of a noncommutative coherent algebra to its underlying scheme. Additionally, we establish that the Krull dimension of a variety over a field is a lower bound for the Rouquier dimension of the bounded derived category associated with a noncommutative coherent algebra on it. This is an extension of a classical result of Rouquier to the noncommutative context.

A gap in the proof of the main result (Theorem 1.3) in our paper [Bull. Lond. Math. Soc. 57 (2025), 79–95] is identified and fixed. The gap is related to possible non‐orientability of the line bundles defined by eigendirections of the (1,1)‐tensor .

We obtain a totient analogue for Linnik's theorem in arithmetic progressions. Specifically, for any coprime pair of positive integers such that is odd, there exists such that .

We study geometric invariant theory for over characteristic zero. For a ‐action on a variety in suitable case, we provide an equivariant birational modification such that is a principal bundle. The situation covers quotients of unstable strata of any linear ‐action.