arXiv subjects
Publications and source records attributed to David Holmes.
We define a new theory of logarithmic Gromov--Witten invariants allowing negative contact orders using pierced logarithmic curves. We prove that these classes satisfy logarithmic birational invariance and satisfy loop and splitting theorems generalising those for classical Gromov--Witten invariants.
The logarithmic double ramification cycle is the virtual fundamental class of the locus where a line bundle on a family of curves is fiberwise trivial. We construct a K-theoretic logarithmic double ramification class and prove a product formula and a \(\mathrm{GL}_r(\mathbb Z)\)-invariance property. We also give an explicit formula for this class in terms of a Grothendieck polynomial via a novel $K$-theoretic Thom--Porteous formula for vector bundles on algebraic stacks.
Large language models (LLMs) have shown increasing promise in solving open problems in mathematics. However, their performance can be further improved through agentic workflows tailored to real-world mathematical practice. To this end, we introduce ProofCouncil, a mathematical agent that is designed to tackle open problems using an author-critic architecture. ProofCouncil served as a submission to the second batch of FirstProof, a challenge consisting of 10 real-world mathematical problems that agents must solve autonomously. Its submissions for 6 of the 10 problems were judged by the referees to be correct up to at most minor revisions, showing the best performance among participating teams. We also evaluate ProofCouncil on 30 open problems collected from mathematical researchers. Among the 21 solutions that received human feedback, 5 were judged completely correct, 2 more were judged promising pending final verification, and a further 8 contained useful partial progress. In this short paper, we describe the development of ProofCouncil and the agent-building library used to create it, which we release as open source to the community.
We study how the effort and success probability of frontier AI systems scale with human difficulty on problems from Project Euler, an online platform of computational mathematics problems. Our dataset, from the MathArena benchmark, consists of 3840 attempts across 50 problems and 26 model configurations, with problem difficulty measured by the site's public human solve times. Motivated by a proposal of Timothy Gowers, we test a power-law relation $t_{\text{machine}} = a \cdot t_{\text{human}}^b$ between generated-token cost per successful answer and human time, and find $b < 1$ for 20 of the 25 models with usable fits, including the strongest base models; this operationalization therefore does not support an earlier prediction that machines scale worse than humans with difficulty. We also investigate whether success probability on the tested problems can be modeled by a simple exponential decay $p_{\text{success}} = e^{c t_{\text{human}}}$, predicting a linear relation between $\log p_{\text{success}}$ and $t_{\text{human}}$. Using a binning approach for data aggregation we find moderate empirical support (median bin-level $R^2 = 0.92$ across the 22 best-covered configurations) for this model. Following METR, we also fit logistic success curves and extract 50\% task-length horizons $h_{50}$; the strongest configurations in our 20 April 2026 snapshot reach roughly $2.5$--$4.3$ hours on our fastest-five human baseline, with a log-linear fit through the state-of-the-art frontier giving a descriptive doubling time of about $75$~days for the SOTA $h_{50}$.
Existing Digital Twin (DT) approaches often lack semantic reasoning capabilities for effective cybersecurity modelling in Cyber-Physical Systems (CPS). This paper presents HySecTwin, a knowledge-driven digital twin architecture that places automated reasoning at the core of real-time threat detection. HySecTwin incorporates semantic modelling to transform heterogeneous CPS telemetry, device attributes, and operational relationships into machine-interpretable representations, combined with an embedded reasoning engine operating over contextualized system states. Unlike opaque detection methods, the framework integrates deterministic rule-based inference with hybrid fuzzy reasoning to generate explicit, interpretable, and auditable security assessments from live device telemetry. This enables context-aware monitoring of complex CPS environments while preserving transparency and trust. Experimental evaluation using a representative CPS testbed and MITRE ATT\&CK campaign-inspired attack scenarios demonstrates sub-millisecond twin synchronization latency and up to 21.5\% faster threat detection compared with deterministic reasoning alone. The results show that semantic modelling, semantic enrichment, and hybrid reasoning improve explainability and resilience without extra system overhead. HySecTwin provides a lightweight, containerized, and extensible framework for secure-by-design digital twin deployments in mission-critical infrastructures
A theta divisor on the universal principally polarised abelian variety can be extended to a compactification either by taking the Zariski closure, or by taking the unique extension which is pure of weight 2. For the latter, following ideas of Yuan and Zhang, we need to pass to the category of adelic- or b-divisors. We show that the two choices of extension differ by a tropicalisation of the Riemann theta function. We prove an extension of Moret-Bailly's ''key formula'' that features the pure weight 2 extension of the theta divisor, and discuss various arithmetic applications, including a ''universal'' formula for the N\'eron--Tate height of a point. A key technical input is the systematic use of the theory of logarithmic abelian varieties due to Kajiwara, Kato, and Nakayama.
As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.
We study classes of strata of differentials with fixed spin parity in the Chow ring of moduli spaces of curves. We show that these classes are tautological and computable. Furthermore, we establish the refined DR cycle formula for these classes.
We define a Grothendieck ring of varieties for log schemes. It is generated by one additional class ``$P$'' over the usual Grothendieck ring. We show the na\"ive definition of log Hodge numbers does not make sense for all log schemes. We offer an alternative that does.
The double ramification (DR) cycle associated to a line bundle on a family of curves detects where the line bundle becomes fibrewise-trivial. The Hodge-DR Conjecture proposes a formula for powers of the first Chern class of a natural line bundle on the DR cycle, with a number of applications in the computation of Euler characteristics of strata of differentials. In this paper we prove the conjecture, as well as an analogue for the logarithmic DR cycle. The proof of the former proceeds via reduction to a localisation computation of Fan, Wu and You; the proof of the latter is based on the Thom--Porteous formula, and as a special case gives a shorter proof of a recent result of Holmes, Molcho, Pandharipade, Pixton and Schmitt. Along the way we develop an analogue of Mumford's formula for the Chern character of the universal line bundle on the universal jacobian over the moduli space of twisted curves, generalising work of Mumford, Chiodo, and Pagani--Ricolfi--van Zelm.
Digital Twins (DTs), optimize operations and monitor performance in Smart Critical Systems (SCS) domains like smart grids and manufacturing. DT-based cybersecurity solutions are in their infancy, lacking a unified strategy to overcome challenges spanning next three to five decades. These challenges include reliable data accessibility from Cyber-Physical Systems (CPS), operating in unpredictable environments. Reliable data sources are pivotal for intelligent cybersecurity operations aided with underlying modeling capabilities across the SCS lifecycle, necessitating a DT. To address these challenges, we propose Security Digital Twins (SDTs) collecting realtime data from CPS, requiring the Shift Left and Shift Right (SLSR) design paradigm for SDT to implement both design time and runtime cybersecurity operations. Incorporating virtual CPS components (VC) in Cloud/Edge, data fusion to SDT models is enabled with high reliability, providing threat insights and enhancing cyber resilience. VC-enabled SDT ensures accurate data feeds for security monitoring for both design and runtime. This design paradigm shift propagates innovative SDT modeling and analytics for securing future critical systems. This vision paper outlines intelligent SDT design through innovative techniques, exploring hybrid intelligence with data-driven and rule-based semantic SDT models. Various operational use cases are discussed for securing smart critical systems through underlying modeling and analytics capabilities.
We construct a derived pushforward of the r-th root of the universal line bundle over the Picard stack of genus g prestable curves carrying a line bundle. We prove a number of basic properties, and give a formula in terms of standard tautological generators. After pullback, our formula recovers formulae of Mumford, of the first-named author, and of Pagani--Ricolfi--van Zelm. We apply these constructions to prove a conjecture expressing the coefficients of higher powers of r in the so-called `Chiodo classes' to the double ramification cycle, and to give a formula for the r-spin logarithmic double ramification cycle.
We introduce a new logarithmic structure on the moduli stack of stable curves, admitting logarithmic gluing maps. Using this we define cohomological field theories taking values in the logarithmic Chow cohomology ring, a refinement of the usual notion of a cohomological field theory. We realise the double ramification cycle as a partial logarithmic cohomological field theory.
We consider functions from the real numbers to the real numbers, output by a neural network with 1 hidden activation layer, arbitrary width, and ReLU activation function. We assume that the parameters of the neural network are chosen uniformly at random with respect to various probability distributions, and compute the expected distribution of the points of non-linearity. We use these results to explain why the network may be biased towards outputting functions with simpler geometry, and why certain functions with low information-theoretic complexity are nonetheless hard for a neural network to approximate.
Multi-scale differentials were constructed by M.~Bainbridge, D.~Chen, Q.~Gendron, S.~Grushevsky, and M.~M\"oller, from the viewpoint of flat and complex geometry, for the purpose of compactifying moduli spaces of curves together with a differential with prescribed orders of zeros and poles. Logarithmic differentials were constructed by S.~Marcus and J.~Wise, as a generalization of stable rubber maps from Gromov--Witten theory. Modulo the global residue condition that isolates the main components of the compactification, we show that these two kinds of differentials are equivalent, and establish an isomorphism of their (coarse) moduli stacks. Moreover, we describe the rubber and multi-scale spaces as an explicit blowup of the moduli space of stable pointed rational curves in the case of genus zero, and as a global blowup of the incidence variety compactification for arbitrary genera, which implies their projectivity. We also propose a refined double ramification cycle formula in the twisted Hodge bundle which interacts with the universal line bundle class.
We show that the ring of Siegel-Jacobi forms of fixed degree and of fixed or bounded ratio between weight and index is not finitely generated. Our main tool is the theory of toroidal b-divisors and their relation to convex geometry. As a byproduct of our methods, we prove a conjecture of Kramer about the representation of all Siegel-Jacobi forms as sections of certain line bundles and we recover a formula due to Tai for the asymptotic dimension of the space of Siegel-Jacobi forms of given ratio between weight and index.
We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and logarithmic jacobians (recently constructed by Molcho and Wise). Our moduli space carries a `double ramification cycle' measuring the locus where the given root is isomorphic to the trivial bundle, and we give a tautological formula for this class in the language of piecewise polynomial functions (as recently developed by Molcho-Pandharipande-Schmitt and Holmes-Schwarz).
We show a Chern-Weil type statement and a Hilbert-Samuel formula for a large class of singular plurisubharmonic metrics on a line bundle over a smooth projective complex variety. For this we use the theory of b-divisors and the so-called multiplier ideal volume function. We apply our results to the line bundle of Siegel-Jacobi forms over the universal abelian variety endowed with its canonical invariant metric. This generalizes a result of the second author for the universal family of elliptic curves to higher degrees.