arXiv subjects

Daniel Holmes

Publications and source records attributed to Daniel Holmes.

7 recordsLinked to original sources

arXiv2026

Experiments with membranes, maps and sheaves

Motivated by the conjectural existence of M-theory, we investigate a web of correspondences between enumerative invariants of a Calabi-Yau fivefold $Z$ with a torus action. This includes a correspondence between the fivefold Gromov-Witten invariants and K-theoretic Pandharipande-Thomas invariants of a threefold $X \subset Z$, mediated by so-called membrane indices, which generalise Gopakumar-Vafa invariants. We establish the correspondence for strip geometries for restricted torus actions and test it numerically for general torus actions. Further numerical evidence is provided for the closed vertex, local surfaces and local projective spaces. For the latter we present conjectural formulae for their low-degree membrane indices. When the fivefold is the product of the affine plane with a suitable toric variety, we prove geometric engineering and equate the generating series of Pandharipande-Thomas invariants with the corresponding instanton partition function while equality with the Gromov-Witten series is probed numerically.

math.AG↗

arXiv2026

Torelli loci, product cycles, and the homomorphism conjecture for $\mathcal{A}_g$

The tautological $\mathbb{Q}$-subalgebra $\mathsf{R}^*(\mathcal{A}_g) \subset \mathsf{CH}^*(\mathcal{A}_g)$ of the Chow ring of the moduli space of principally polarized abelian varieties is generated by the Chern classes of the Hodge bundle. There is a canonical $\mathbb{Q}$-linear projection operator $\mathsf{taut}: \mathsf{CH}^*(\mathcal{A}_g) \rightarrow \mathsf{R}^*(\mathcal{A}_g).$ We present here new calculations of intersection products of the Torelli locus in $\mathcal{A}_g$ with the product loci $\mathcal{A}_{r}\times \mathcal{A}_{g-r} \rightarrow \mathcal{A}_g$ for $r\leq 3$. The results suggest that $\mathsf{taut}$ is a $\mathbb{Q}$-algebra homomorphism, at least for special cycles. We discuss a conjectural framework for this homomorphism property. Our calculations follow two independent approaches. The first is a direct study of the excess intersection geometry of the fiber product of the Torelli and product morphisms. The second recasts the geometry in terms of families Gromov-Witten classes, which are computed by a wall-crossing formula related to unramified maps. We define tautological projections of cycles on the fiber products $\mathcal X_g^s \to \mathcal A_g$ of the universal family. We compute these projections for a class of product cycles on $\mathcal X_g^s$ in terms of a determinant involving the universal theta divisors and Poincar\'e classes. Using Abel-Jacobi pullbacks of product cycles on $\mathcal X_g^s$ and their projections, we construct a new family of classes which we conjecture to lie in the Gorenstein kernels of the tautological rings $\mathsf{R}^*(\mathcal M^{\mathrm{ct}}_{g,n})$. In particular, we construct nontrivial elements of the Gorenstein kernels of $\mathsf{R}^5(\mathcal{M}_{5,2}^{\mathrm{ct}})$ and $\mathsf{R}^5(\mathcal{M}_{4,4}^{\mathrm{ct}})$.

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

Computations in equivariant Gromov-Witten theory of GKM spaces

We study equivariant Gromov-Witten invariants and quantum cohomology in GKM theory. Building on the localization formula, we prove that the resulting expression is independent of the choice of compatible connection, and provide an equivalent formulation without auxiliary choices. Motivated by this theoretical refinement, we develop a software package, $\mathtt{GKMtools . jl}$, that implements the computation of equivariant GW invariants and quantum products directly from the GKM graph. We apply our framework to several geometric settings: Calabi-Yau rank two vector bundles on the projective line, where we obtain a new proof for a recent connection to Donaldson-Thomas theory of Kronecker quivers; twisted flag manifolds, which give symplectic but non-algebraic examples of GKM spaces; realizability questions for abstract GKM graphs; classical enumerative problems involving curves in hyperplanes; and quantum Schubert calculus for smooth Schubert varieties. These results demonstrate both the theoretical flexibility of GKM methods and the effectiveness of computational tools in exploring new phenomena.

math.AG↗

arXiv2024

Design Principles of Dynamic Resource Management for High-Performance Parallel Programming Models

With Dynamic Resource Management (DRM) the resources assigned to a job can be changed dynamically during its execution. From the system's perspective, DRM opens a new level of flexibility in resource allocation and job scheduling and therefore has the potential to improve system efficiency metrics such as the utilization rate, job throughput, energy efficiency, and responsiveness. From the application perspective, users can tailor the resources they request to their needs offering potential optimizations in queuing time or charged costs. Despite these obvious advantages and many attempts over the last decade to establish DRM in HPC, it remains a concept discussed in academia rather than being successfully deployed on production systems. This stems from the fact that support for DRM requires changes in all the layers of the HPC system software stack including applications, programming models, process managers, and resource management software, as well as an extensive and holistic co-design process to establish new techniques and policies for scheduling and resource optimization. In this work, we therefore start with the assumption that resources are accessible by processes executed either on them (e.g., on CPU) or controlling them (e.g., GPU-offloading). Then, the overall DRM problem can be decomposed into dynamic process management (DPM) and dynamic resource mapping or allocation (DRA). The former determines which processes (or which change in processes) must be managed and the latter identifies the resources where they will be executed. The interfaces for such \mbox{DPM/DPA} in these layers need to be standardized, which requires a careful design to be interoperable while providing high flexibility. Based on a survey of existing approaches we propose design principles, that form the basis of a holistic approach to DMR in HPC and provide a prototype implementation using MPI.

cs.DC↗

arXiv2021

Affine dimers from characteristic polygons

Recent work by Forsg{\aa}rd indicates that not every convex lattice polygon arises as the characteristic polygon of an affine dimer or, equivalently, an admissible oriented line arrangement on the torus in general position. We begin the classification of convex lattice polygons arising as characteristic polygons of affine dimers. We present several general constructions of new affine dimers from old, and an algorithm for finding affine dimers with prescribed polygon. With these tools we prove that all lattice triangles, generalised parallelograms, and polygons of genus at most two admit an affine dimer.

math.GT↗

arXiv2019

Extending the Message Passing Interface (MPI) with User-Level Schedules

Composability is one of seven reasons for the long-standing and continuing success of MPI. Extending MPI by composing its operations with user-level operations provides useful integration with the progress engine and completion notification methods of MPI. However, the existing extensibility mechanism in MPI (generalized requests) is not widely utilized and has significant drawbacks. MPI can be generalized via scheduled communication primitives, for example, by utilizing implementation techniques from existing MPI-3 nonblocking collectives and from forthcoming MPI-4 persistent and partitioned APIs. Non-trivial schedules are used internally in some MPI libraries; but, they are not accessible to end-users. Message-based communication patterns can be built as libraries on top of MPI. Such libraries can have comparable implementation maturity and potentially higher performance than MPI library code, but do not require intimate knowledge of the MPI implementation. Libraries can provide performance-portable interfaces that cross MPI implementation boundaries. The ability to compose additional user-defined operations using the same progress engine benefits all kinds of general purpose HPC libraries. We propose a definition for MPI schedules: a user-level programming model suitable for creating persistent collective communication composed with new application-specific sequences of user-defined operations managed by MPI and fully integrated with MPI progress and completion notification. The API proposed offers a path to standardization for extensible communication schedules involving user-defined operations. Our approach has the potential to introduce event-driven programming into MPI (beyond the tools interface), although connecting schedules with events comprises future work. Early performance results described here are promising and indicate strong overlap potential.

cs.DC↗