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Martin Möller

Publications and source records attributed to Martin Möller.

At least 19 recordsLinked to original sources
arXiv2026

LLM-Assisted Review Prioritization for German Statutory Health Insurance Websites: A Multi-Stage Corpus Audit

Background: German statutory health insurance (SHI) funds publish web portfolios that exceed continuous specialist review capacity. Their content can shape health and benefit expectations. Generic AI-text detection does not identify medical, benefit, legal, or editorial review needs. Objective: To characterize a multi-stage workflow that prioritizes substantive review needs while separating AI-provenance signals from quality claims. Methods: We analyzed 56,198 pages from 84 SHI websites or sub-sites. The workflow combined deterministic screening, model-assisted triage and in-depth review, minimum evidence checks, temporal-validity safeguards, and paired-model comparison. It is reproducibility-bounded, not a validated detector. Production code is proprietary; reproducibility rests on frozen derived tables and paired-comparison artifacts. The 300-page lower-priority check was a single-model, risk-enriched routing stress test, not a human-reference evaluation. Results: All pages received a review state. The workflow generated 35,998 review records and routed 21,452 to case review. The workload concentrated in transparency, legal framing, medical content, contradictions, and AI-related failure-mode signals. A quoted passage was locatable in captured page text for 31,347 records, confirming literal occurrence rather than factual correctness. The routing stress test surfaced a signal on 100/300 pages (33.3% within the sample). Across 182 matched cases, two models agreed in 75.8% (kappa = 0.532; 95% CI 0.415-0.649). Conclusions: The workflow produces a prioritized workload, not error prevalence or final legal, medical, or insurer-level findings. It neither proves AI authorship nor validates autonomous detection. Paired-model agreement quantifies consistency, not correctness or sufficient triage performance; public claims require human adjudication.

cs.CY↗
arXiv2026

Completed volumes and the DR-cycle

We show that the completed volumes introduced by Duriev-Goujard-Yakovlev as an approximation to compute Masur-Veech volumes via Witten-Kontsevich's combinatorial classes agrees with the top intersection of the tautological class on the double ramification cycle, computable as a coefficient of a Chiodo class. For the proof we describe the components of the double ramification cycle and their excess intersection classes to the extent seen by the top tautological intersection. This gives a recursion computing completed volumes in terms of volumes appearing in a certain set of level graphs, not only for quadratic differentials. It also completes the work of Duriev-Goujard-Yakovlev solving the technically most involved case of strata with two singularities.

math.AG↗
arXiv2026

Shaping the Digital Future of ErUM Research: Sustainability & Ethics

This workshop report from "Shaping the Digital Future of ErUM Research: Sustainability & Ethics" (Aachen, 2025) reviews progress on sustainability measures in data-intensive ErUM-Data research since the 2023 call-to-action on resource-aware research. It evaluates short-, medium-, and long-term actions around monitoring and reducing CO2 emissions, improving data and software FAIRness, optimizing workflows and computing infrastructures, and aligning operations with low-carbon energy availability, including concepts such as "breathing" computing centers, long-term data storage strategies, and software efficiency certification. The report stresses the need for systematic teaching, training, mentoring, and new support formats to establish sustainable coding and computing practices, particularly among students and early-career researchers, and highlights the importance of dedicated steering and funding instruments to embed sustainability in project planning. Ethical discussions focus on the transformative use of AI in ErUM-Data, addressing autonomy, bias, transparency, explainability, attribution of responsibility, and the risk of deskilling, while reaffirming that accountability for scientific outcomes remains with human researchers. Finally, the report emphasizes that sustainable transformation requires not only technical measures but also targeted awareness-building, communication strategies, incentives, and community-driven initiatives to move from awareness to action and to integrate sustainability and ethics into everyday scientific practice.

physics.comp-ph↗
arXiv2026

Intersections of special cycles on Shimura curves and Siegel Maass forms

We show that the generating series of the number of pairs of geodesics on a compact Shimura curve with given discriminants and intersection angle are coefficients of a non-holomorphic Siegel modular form, a theta lift of the constant function. This retrieves and generalizes counting results of Rickards via the Siegel-Weil formula. More generally, we study the genus two theta lift of Maass forms on this Shimura curve and prove a Fourier-Taylor expansion in terms of some generalized Whittaker functions. We also provide a geometric interpretation of all Fourier coefficients of these theta lifts in terms of averages of geodesic Taylor coefficients over special cycles.

math.NT↗
arXiv2025

The Kodaira dimension of even spin strata of Abelian differentials

The even spin components of the strata of Abelian differentials are difficult to handle from a birational geometry perspective due to the fact that their spin line bundles have more sections than expected. Nevertheless, in this paper, we prove that for large genus, the minimal even spin components are of general type. This result complements the previous work by the second and third authors, together with Costantini, on the Kodaira dimension of general strata and the minimal odd spin components of Abelian differentials. Our main technical tool is the computation and estimation of a series of effective divisor classes on the even spin components.

math.AG↗
arXiv2025

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

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.

cs.CL↗
arXiv2025

Wall-crossing formulas via spectral networks

We give a self-contained proof of the Kontsevich-Soibelman wall-crossing formula entirely in the scope of quadratic differentials without relying on input from DT theory. Our approach is based on path-lifting rules for spectral networks introduced by Gaiotto, Moore and Neitzke. We provide a framework to justify the convergence of the path liftings, including the cases with spiral domains. In particular, we define path lifting rules for spectral networks associated to holomorphic quadratic differentials. As an intermediate step in the proof of the wall-crossing formula, we show that upon extending the path lifting rules to $\mathcal{A}_0$-laminations we generate the hat-homology lattice.

math.AG↗
arXiv2025

Area Siegel--Veech constants for affine invariant submanifolds of REL zero

We describe the principal boundary of an arbitrary affine invariant submanifold of REL zero in terms of level graphs of the multi-scale compactification of strata of Abelian differentials with prescribed orders of zeros. We show that the area Siegel--Veech constant of the affine invariant submanifold can be obtained by using volumes of the principal boundary strata. As an application, we prove the conjectural formula in [CMS23a] that computes the area Siegel--Veech constant via intersection theory in the case of REL zero. In particular, the formula holds for strata of quadratic differentials with odd orders of zeros and for the gothic locus. We also explicitly describe the principal boundary components of the gothic locus and their individual contributions to the area Siegel--Veech constant

math.GT↗
arXiv2025

Chern classes of linear submanifolds with application to spaces of k-differentials and ball quotients

We provide formulas for the Chern classes of linear submanifolds of the moduli spaces of Abelian differentials and hence for their Euler characteristic. This includes as special case the moduli spaces of k-differentials, for which we set up the full intersection theory package and implement it in the sage-program diffstrata. As an application, we give an algebraic proof of the theorems of Deligne-Mostow and Thurston that suitable compactifications of moduli spaces of k-differentials on the 5-punctured projective line with weights satisfying the INT-condition are quotients of the complex two-ball.

math.AG↗
arXiv2024

The moduli space of multi-scale differentials

We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with normal crossing boundary. Locally, our compactification can be described as the normalization of an explicit blowup of the incidence variety compactification, which was defined in [BCGGM18] as the closure of the stratum of abelian differentials in the closure of the Hodge bundle. We also define families of projectivized multi-scale differentials, which gives a proper Deligne-Mumford stack, and our compactification is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized moduli space of multi-scale differentials such that the $\mathrm{GL}_2(\mathbb R)$-action in the interior of the moduli space extends continuously to the boundary.

math.AG↗
arXiv2024

A smooth compactification of spaces of stability conditions: the case of the $A_{n}$-quiver

We propose a notion of multi-scale stability conditions with the goal of providing a smooth compactification of the quotient of the space of projectivized Bridgeland stability conditions by the group of autoequivalence. For the case of the 3CY category associated with the $A_n$-quiver this goal is achieved by defining a topology and complex structure that relies on a plumbing construction. We compare this compactification to the multi-scale compactification of quadratic differentials and briefly indicate why even for the Kronecker quiver this notion needs refinement to provide a full compactification.

math.AG↗
arXiv2024

Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori

Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel-Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemannian surfaces. This space carries actions of the foliated Laplacian derived from the SL(2,R)-action as well as various differential operators related to relative period translations. In the paper we give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows to employ techniques from representation theory and global analysis. Even for this simple stratum exhibiting all Siegel-Veech transforms requires novel configurations of saddle connections. We also show that the contiunuous spectrum of the foliated Laplacian is much larger than the space of Siegel-Veech transforms, as opposed to the case of the modular curve. This defect can be remedied by using instead a compound Laplacian involving relative period translations.

math.NT↗
arXiv2024

Quadratic differentials as stability conditions: collapsing subsurfaces

We introduce a new class of triangulated categories, which are Verdier quotients of three-Calabi-Yau categories from (decorated) marked surfaces, and show that its spaces of stability conditions can be identified with moduli spaces of framed quadratic differentials on Riemann surfaces with arbitrary order zeros and arbitrary higher order poles. A main tool in our proof is a comparison of two exchange graphs, obtained by tilting hearts in the quotient categories and by flipping mixed angulations associated with the quadratic differentials.

math.GT↗
arXiv2024

Compactifying the rank two Hitchin system via spectral data on semistable curves

We study resolutions of the rational map to the moduli space of stable curves that associates with a point in the Hitchin base the spectral curve. In the rank two case the answer is given in terms of the space of quadratic multi-scale differentials introduced in [BCGGM3]. This space defines a compactification (of the projectivization) of the regular locus of the $\mathrm{GL}(2,\mathbb{C})$-Hitchin base and provides a compactification of the Hitchin system by compactified Jacobians of pointed stable curves. We show how the classical $\mathrm{GL}(2,\mathbb{C})$- and $\mathrm{SL}(2,\mathbb{C})$-spectral correspondence extend to the compactified Hitchin system by a correspondence along an admissible cover between torsion-free rank $1$ sheaves and (multi-scale) Higgs pairs of rank $2$.

math.AG↗
arXiv2023

Teichmüller curves in hyperelliptic components of meromorphic strata

We provide a complete classification of Teichmüller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichmüller curves intersect the boundary of the moduli space we derive a contradiction to the algebraicity of any candidate outside of Hurwitz covers of strata with projective dimension one and Hurwitz covers of zero residue loci in strata with projective dimension two.

math.AG↗
arXiv2022

A tale of two moduli spaces: logarithmic and multi-scale differentials

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.

math.AG↗
arXiv2022

On the Kodaira dimension of moduli spaces of Abelian differentials

This paper lays the foundation for determining the Kodaira dimension of the projectivized strata of Abelian differentials with prescribed zero and pole orders in large genus. We work with the moduli space of multi-scale differentials constructed in [BCGGM2] which provides an orbifold compactification of these strata. We establish the projectivity of the moduli space of multi-scale differentials, describe the locus of canonical singularities, and compute a series of effective divisor classes. Moreover, we exhibit a perturbation of the canonical class which allows the corresponding pluri-canonical forms to extend over the locus of non-canonical singularities. As applications, we certify general type for strata with few zeros as well as for strata with equidistributed zero orders when g is sufficiently large. In particular, we show general type for the odd spin components of the minimal strata for g > 12.

math.AG↗