Royal Society

We prove that the probability the cluster of the origin in a subcritical Poisson random connection model (RCM) has size at least n decays exponentially as n increases, under minimal assumptions. We extend a recent method of Vanneuville from Bernoulli percolation on vertex-transitive graphs to the RCM, where the vertex set has little structure. The key idea is that the subcritical RCM can be constructed by site percolation on a high-intensity RCM. The latter RCM becomes “almost vertex-transitive” at very high intensities. We obtain the result for connection functions with unbounded support, a setting in which it was not previously known.

Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum mechanics to reformulations of the Einstein equations and network theory. Motivated by advances in compatible and structure-preserving discretisation such as Finite Element Exterior Calculus (FEEC), we examine how differential complexes encode critical properties such as existence, uniqueness, stability and rigidity of solutions to differential equations. We demonstrate that various fundamental concepts and models in solid and fluid mechanics are essentially formulated in terms of differential complexes.

Biodiversity is declining in many parts of the world. Biological diversity measurement and monitoring are fundamental to the assessment of the causes and consequences of environmental changes, identification of key areas for the protection of biodiversity or ecosystem services, determining the effectiveness of actions, and the creation of decision-support tools critical to maintaining a sustainable planet. Biodiversity measurement is rapidly changing due to advances in citizen science, image recognition, acoustic monitoring, environmental DNA, genomics, remote sensing, and AI. In this perspective, we outline the exciting opportunities these developments offer but also consider the challenges. Our key recommendations are to 1) Capitalize on the ability of novel technology to integrate data sources 2) agree to standard methods for data collection 3) ensure new technologies are calibrated with existing data; 4) fill data gaps by using emerging technologies and increasing capacity, especially in the tropics; 5) create living safeguarded databases of trusted information to reduce the risk of poisoning by AI hallucinated, or false, information; 6) ensure data generation is valued; 7) ensure respectful incorporation of Indigenous Knowledge; 8) ensure measurements enable the quantification of effectiveness of actions, and 9) increase the resilience of global datasets to technical and societal change. Radical new collaborations are needed between computer scientists, engineers, molecular biologists, data scientists, field ecologists, citizen scientists, Indigenous peoples, policymakers, and local communities to create the rigorous, resilient, accessible biodiversity information systems required to underpin policies and practices that ensure the maintenance and restoration of ecological systems.

Let G be a connected, semisimple, simply connected algebraic group over an algebraically closed field of positive characteristic. For each restricted dominant weight λ, there is the associated principal indecomposable G_1-module Q_1(λ), where G_1 is the first infinitesimal subgroup of G. The assertion that, for every such λ, there exists a G-module whose restriction to G_1 is isomorphic to Q_1(λ) is known as the Humphreys–Verma Conjecture. For groups of rank 2, it was shown in that the Humphreys–Verma Conjecture holds in all cases except one, namely when G is of type G_2, the characteristic is 2, and λ=0. This case remained completely open. Moreover, in every previously resolved case, the module Q_1(λ) could be realized as the restriction of a suitable tilting module. However, in it was shown that Q_1(0) for G_2 in characteristic 2 cannot arise as the restriction of a tilting module, thereby providing the first counterexample to a conjecture of the first author. In this paper, we construct a G-module whose restriction to G_1 is Q_1(0), thereby establishing the Humphreys–Verma Conjecture in the last remaining rank 2 case. Our construction provides the first known example of a G-structure on a principal indecomposable G_1-module that does not arise from a tilting module. This reveals a new phenomenon in the study of the Humphreys–Verma Conjecture and suggests new directions for understanding G-structures on principal indecomposable G_1-modules.

We prove that mutation-acyclicity of quivers on four vertices is decidable. If a four-vertex quiver Q is mutation-acyclic, then some sequence of mutations takes it to an acyclic quiver without any arrow multiplicity exceeding max{ht(Q),m(Q)}, where ht(Q) is the largest arrow multiplicity of Q and m(Q) is the Markov invariant of Fomin and Neville evaluated on the totally proper wiggle class of Q. A search within a computable form of this bound decides mutation-acyclicity of four-vertex quivers with O(h^24) states for input height h≥1. Together with a theorem of Caldero and Keller, it also decides mutation equivalence of two four-vertex quivers up to relabeling whenever one of them is mutation-acyclic. We conjecture that ht(Q) alone suffices as the bound, which would reduce the search to O(h^6) states, and we verify this for all four-vertex quivers of height at most three. In the other direction, for every n≥4 we construct an affine family of connected n-vertex quivers whose nonnegative even members are mutation-acyclic and whose positive odd members are mutation-cyclic. Consequently, unlike in rank three, no fixed finite Boolean combination of polynomial equations and strict polynomial inequalities in the arrow weights (i.e., a semialgebraic criterion) decides mutation-acyclicity of n-vertex quivers, even allowing polynomials that are not mutation-invariant. Finally, we correct the criterion for four-vertex quivers of height at most two stated in our earlier paper, and recover its classification counts with short certificates. The calculations in the proof of the height bound are verified by computer for all values of their parameters.