We develop an analytic construction of nowhere-vanishing harmonic 1-forms on real loci of K3-fibred Calabi-Yau 3-folds with collapsing Ricci-flat Kähler metrics. We apply our construction to examples whose real loci have connected components diffeomorphic to S^1× S^2 and to both trivial and nontrivial mapping tori. As an application, we produce examples of compact 7-manifold with holonomy G_2 via the Joyce-Karigiannis construction.

We construct Spin(7)-instantons on one of Joyce’s compact Spin(7)-manifolds. The underlying compact Spin(7)-manifold given by Joyce is the same as in Lewis’ construction of Spin(7)-instantons. However, our construction method and the resulting instantons are new. The compact Spin(7)-manifold is constructed by gluing a Spin(7)-orbifold and certain local model spaces around the orbifold singularities. We construct our instantons by gluing non-flat connections on the local model spaces to a flat connection on the Spin(7)-orbifold. We deliver more than 20, 000 new four-parameter families of examples of Spin(7)-instantons within the structure groups SO(3), SO(4), SO(5), SO(7), and SO(8).

We explain a construction of G_2 -instantons on manifolds obtained by resolving G_2 -orbifolds. This includes the case of G_2 -instantons on resolutions of T^7/Γ as a special case. The ingredients needed are a G_2 -instanton on the orbifold and a Fueter section over the singular set of the orbifold which are used in a gluing construction. In the general case, we make the very restrictive assumption that the Fueter section is pointwise rigid. In the special case of resolutions of T^7/Γ , improved control over the torsion-free G_2 -structure allows to remove this assumption. As an application, we construct a large number of G_2 -instantons on the simplest example of a resolution of T^7/Γ . We also construct one new example of a G_2 -instanton on the resolution of (T^3 ×K3)/ℤ^2_2 .

We use supervised machine learning to predict Hodge numbers for Calabi-Yau threefolds encoded by reflexive polyhedra. The Hodge number is invariant to the order of the vertices and the swapping of axes. Incorporating these properties, i.e. the invariance of column and row permutations for a matrix containing the polyhedron's vertices, promises better performance for Hodge number prediction. On a medium-sized subset of the KreuzerSkarke dataset, we train and evaluate approaches with different degrees of invariance. Our comparison shows that machine learning models incorporating symmetries actually outperform models that do not, with our best model achieving almost 97% accuracy.

Public and permissionless blockchain systems are challenged by Sybil attacks, in which attackers use multiple identities to gain control. Traditionally, such attacks are prevented by consensus mechanisms relying on resource expenditure. However, such mechanisms (e.g. proof of work) face criticism for being wasteful. To address this and other concerns, novel blockchain systems backed by new consensus mechanisms have recently emerged. We formalise three key characteristics pursued by these systems: permissionlessness, Sybil attack resistance, and freeness. We demonstrate that no blockchain protocol can simultaneously achieve all three characteristics within the paradigm established by our formalisation. Thus, a trilemma emerges for distributed ledger technology designers, who must balance these characteristics thoughtfully. [GRAPHICS]

Let X be a complex algebraic K3 surface of degree 2d and with Picard number ρ . Assume that X admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, ρ≥ 1 when d=1 and ρ≥ 2 when d ≥ 2 . For d=1 , the first example defined over ℚ with ρ =1 was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kondō, also defined over ℚ , can be used to realise the minimum ρ =2 for all d≥ 2 . In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum ρ =2 for d=2,3,4 . We also show that a nodal quartic surface can be used to realise the minimum ρ =2 for infinitely many different values of d. Finally, we strengthen a result of Morrison by showing that for any even lattice N of rank 1≤ r ≤ 10 and signature (1,r-1) there exists a K3 surface Y defined over ℝ such that Pic Y_ℂ= Pic Y ≅ N .

We numerically study whether there exist nowhere vanishing harmonic $1$-forms on the real locus of some carefully constructed examples of Calabi-Yau manifolds, which would then give rise to potentially new examples of $G_2$-manifolds and an explicit description of their metrics. We do this in two steps: first, we use a neural network to compute an approximate Calabi-Yau metric on each manifold. Second, we use another neural network to compute an approximately harmonic $1$-form with respect to the approximate metric, and then inspect the found solution. On two manifolds existence of a nowhere vanishing harmonic $1$-form can be ruled out using differential geometry. The real locus of a third manifold is diffeomorphic to $S^1 \times S^2$, and our numerics suggest that when the Calabi-Yau metric is close to a singular limit, then it admits a nowhere vanishing harmonic $1$-form. We explain how such an approximate solution could potentially be used in a numerically verified proof for the fact that our example manifold must admit a nowhere vanishing harmonic $1$-form.

We explain a construction of G(2)-instantons on manifolds obtained by resolving G(2)-orbifolds. This includes the case of G(2)-instantons on resolutions of T-7/Gamma as a special case. The ingredients needed are a G(2)-instanton on the orbifold and a Fuetersection over the singular set of the orbifold which are used in a gluing construction. In the general case, we make the very restrictive assumption that the Fueter section is point wise rigid. In the special case of resolutions of T-7/Gamma, improved control over the torsion-free G(2)-structure allows to remove this assumption. As an application, we construct a large number of G(2)-instantons on the simplest example of a resolution of T-7/Gamma.Wealsoconstruct one new example of a G(2)-instanton on the resolution of (T-3 x K-3)/Z(2)(2).

Generative artificial intelligence (GenAI) in general, and large language models (LLMs) in particular, are highly fashionable.As they have the ability to generate coherent output based on prompts in natural language, they are promoted as tools to free knowledge workers from tedious tasks such as content writing, customer support and routine computer code generation.Unsurprisingly, their application is also attractive to professionals in the research domain, where mundane and laborious tasks, such as literature screening, are commonplace.We evaluate Vertex AI 'text-bison', a foundational LLM model, in a real-world academic scenario by replicating parts of a popular systematic review in the information management domain.By comparing the results of a zero-shot LLM-based approach with those of the original study, we gather evidence on the suitability of state-of-theart general-purpose LLMs for the analysis of scientific content.We show that the LLM-based approach delivers good scientific content analysis performance for a general classification problem (ACC = 0.9), acceptable performance for a domain-specific classification problem (ACC = 0.8) and borderline performance for a text comprehension problem (ACC ≈ 0.69).We conclude that some content analysis tasks with moderate accuracy requirements may be supported by current LLMs.As the technology will evolve rapidly in the foreseeable future, studies on large corpora, where some inaccuracies are tolerable, or workflows that prepare large data sets for human processing, may increasingly benefit from the capabilities of GenAI.

This article constructs examples of associative submanifolds in $G_2$-manifolds obtained by resolving $G_2$-orbifolds using Joyce's generalised Kummer construction. As the $G_2$-manifolds approach the $G_2$-orbifolds, the volume of the associative submanifolds tends to zero. This partially verifies a prediction due to Halverson and Morrison.

We approach the well-studied problem of supervised group invariant and equivariant machine learning from the point of view of geometric topology. We propose a novel approach using a pre-processing step, which involves projecting the input data into a geometric space which parametrises the orbits of the symmetry group. This new data can then be the input for an arbitrary machine learning model (neural network, random forest, support-vector machine etc). We give an algorithm to compute the geometric projection, which is efficient to implement, and we illustrate our approach on some example machine learning problems (including the well-studied problem of predicting Hodge numbers of CICY matrices), in each case finding an improvement in accuracy versus others in the literature. The geometric topology viewpoint also allows us to give a unified description of so-called intrinsic approaches to group equivariant machine learning, which encompasses many other approaches in the literature.

Popular permissionless distributed ledger technology (DLT) systems using proof-of-work (PoW) for Sybil attack resistance have extreme energy requirements, drawing stern criticism from academia, business and the media. DLT systems building on alternative consensus mechanisms, particularly proof-of-stake (PoS), aim to address this downside. In this paper, we take an initial step towards comparing the energy requirements of such systems to understand whether they achieve this goal equally well. While multiple studies have analysed the energy demands of individual blockchains, little comparative work has been done. We approach this research gap by formalising a basic consumption model for PoS blockchains. Applying this model to six archetypal blockchains generates three main findings. First, we confirm the concerns around the energy footprint of PoW by showing that Bitcoin's energy consumption exceeds the energy consumption of all PoS-based systems analysed by at least three orders of magnitude. Second, we illustrate that there are significant differences in energy consumption among the PoS-based systems analysed, with permissionless systems having a larger energy footprint overall owing to their higher replication factor. Third, we point out that the type of hardware that validators use has a considerable impact on whether the energy consumption of PoS blockchains is comparable with or considerably larger than that of centralised systems.

The resolution of the $G_2$-orbifold $T^7/\Gamma$, where $\Gamma$ is a suitably chosen finite group, admits a $1$-parameter family of $G_2$-structures with small torsion $\varphi^t$, obtained by gluing in Eguchi-Hanson spaces. It was shown by Joyce that $\varphi^t$ can be perturbed to torsion-free $G_2$-structures $\tilde{\varphi}^t$ for small values of $t$. Using norms adapted to the geometry of the manifold we give an alternative proof of the existence of $\tilde{\varphi}^t$. This alternative proof produces the estimate $\left|\left| \tilde{\varphi}^t-\varphi^t \right|\right|_{C^0} \leq ct^{5/2}$. This is an improvement over the previously known estimate $\left|\left| \tilde{\varphi}^t-\varphi^t \right|\right|_{C^0} \leq ct^{1/2}$. As part of the proof, we show that Eguchi-Hanson space admits a unique (up to scaling) harmonic form with decay, which is a result of independent interest.

Popular distributed ledger technology (DLT) systems using proof-of-work (PoW) for Sybil attack resistance have extreme energy requirements, drawing stern criticism from academia, businesses, and the media. DLT systems building on alternative consensus mechanisms, foremost proof-of-stake (PoS), aim to address this downside. In this paper, we take a first step towards comparing the energy requirements of such systems to understand whether they achieve this goal equally well. While multiple studies have been undertaken that analyze the energy demands of individual Blockchains, little comparative work has been done. We approach this research question by formalizing a basic consumption model for PoS blockchains. Applying this model to six archetypal blockchains generates three main findings: First, we confirm the concerns around the energy footprint of PoW by showing that Bitcoin’s energy consumption exceeds the energy consumption of all PoS-based systems analyzed by at least three orders of magnitude. Second, we illustrate that there are significant differences in energy consumption among the PoSbased systems analyzed, with permissionless systems having an overall larger energy footprint. Third, we point out that the type of hardware that validators use has a considerable impact on whether PoS blockchains’ energy consumption is comparable with or considerably larger than that of centralized, non-DLT systems.