arXiv subjects

Jonas Henkel

Publications and source records attributed to Jonas Henkel.

5 recordsLinked to original sources

arXiv2026

Explicit Laplace Spectra of Homogeneous Principal Bundles

We present a unified representation-theoretic method to compute the Laplace-Beltrami spectrum on homogeneous principal bundles. For this setting, we introduce a multi-parameter family of metric deformations called generalized canonical variations. Building upon the geometric realization of such fibrations as naturally reductive spaces, we establish a simplified spectral branching criterion. We apply this method to derive the full spectra (yielding all eigenvalues and multiplicities) for several prominent geometric families. Specifically, we compute the full spectra for the entire classical series of homogeneous 3-$(\alpha,\delta)$-Sasaki manifolds (Types A, B, C, and D) and for real and complex Stiefel manifolds over Grassmannians. These explicit formulas provide the analytical data to investigate related problems in geometric analysis. As an application, we classify the scalar stability of these spaces under Perelman's $\nu$-entropy and, for the 3-$(\alpha,\delta)$-Sasaki manifolds, determine the exact thresholds for Yamabe bifurcations.

math.DG↗

arXiv2026

Hodge Laplacian on $1$-forms of homogeneous $3$-spheres

We study the spectrum of the Hodge-Laplacian on $1$-forms for left-invariant metrics on the Lie group $\operatorname{SU}(2) \cong S^3$ and its quotient $\operatorname{SO}(3)\cong P^3(\mathbb{R})$. To the best of our knowledge, we provide the first explicit computation of the full spectrum of the Hodge-Laplacian for a canonical variation by determining the eigenvalues of Berger 3-spheres and analyzing their resulting splitting behavior. Furthermore, we propose and rigorously prove an explicit formula for the first eigenvalue of general homogeneous metrics on $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$. The formal proof of this result was autonomously discovered by an advanced AI model, providing a notable case study for AI-driven mathematical research. Finally, leveraging this explicit formula, we apply these spectral results to the inverse problem, showing that the spectrum on $1$-forms determines the metric up to isometry. The source code for the symbolic computations, visualizations, and a Monte Carlo stress test is provided in the electronic supplementary material [He26].

math.DG↗

arXiv2026

A Machine Learning Approach to the Nirenberg Problem

This work introduces the Nirenberg Neural Network: a numerical approach to the Nirenberg problem of prescribing Gaussian curvature on $S^2$ for metrics that are pointwise conformal to the round metric. Our mesh-free physics-informed neural network (PINN) approach directly parametrises the conformal factor globally and is trained with a geometry-aware loss enforcing the curvature equation. Additional consistency checks were performed via the Gauss-Bonnet theorem, and spherical-harmonic expansions were fit to the learnt models to provide interpretability. For prescribed curvatures with known realisability, the neural network achieves very low losses ($10^{-7} - 10^{-10}$), while unrealisable curvatures yield significantly higher losses. This distinction enables the assessment of unknown cases, separating likely realisable functions from non-realisable ones. The current capabilities of the Nirenberg Neural Network demonstrate that neural solvers can serve as exploratory tools in geometric analysis, offering a quantitative computational perspective on longstanding existence questions.

cs.LG↗

arXiv2025

The Mathematician's Assistant: Integrating AI into Research Practice

The rapid development of artificial intelligence (AI), marked by breakthroughs like 'AlphaEvolve' and 'Gemini Deep Think', is beginning to offer powerful new tools that have the potential to significantly alter the research practice in many areas of mathematics. This paper explores the current landscape of publicly accessible large language models (LLMs) in a mathematical research context, based on developments up to August 2, 2025. Our analysis of recent benchmarks, such as MathArena and the Open Proof Corpus (Balunovi\'c et al., 2025; Dekoninck et al., 2025), reveals a complex duality: while state-of-the-art models demonstrate strong abilities in solving problems and evaluating proofs, they also exhibit systematic flaws, including a lack of self-critique and a model depending discrepancy between final-answer accuracy and full-proof validity. Based on these findings, we propose a durable framework for integrating AI into the research workflow, centered on the principle of the augmented mathematician. In this model, the AI functions as a copilot under the critical guidance of the human researcher, an approach distilled into five guiding principles for effective and responsible use. We then systematically explore seven fundamental ways AI can be applied across the research lifecycle, from creativity and ideation to the final writing process, demonstrating how these principles translate into concrete practice. We conclude that the primary role of AI is currently augmentation rather than automation. This requires a new skill set focused on strategic prompting, critical verification, and methodological rigor in order to effectively use these powerful tools.

math.HO↗

arXiv2025

The Laplace-Beltrami spectrum on Naturally Reductive Homogeneous Spaces

We prove a formula for the spectrum of the Laplace-Beltrami operator on functions for compact naturally reductive homogeneous spaces in terms of eigenvalues of a generalized Casimir operator and spherical representations. We apply this result to a large family of canonical variations of normal homogeneous metrics, thus allowing for the first time to study how the spectrum depends on the deformation parameters of the metric. As an application, we provide a formula for the full spectrum of compact positive homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds (a family of metrics which includes, in particular, all homogeneous $3$-Sasaki manifolds). The second part of the paper is devoted to the detailed computation and investigation of the spectrum of this family of metrics on the Aloff-Wallach manifold $W^{1,1}=SU(3)/S^{1}$; in particular, we provide a documented Python script that allows the explicit computation in any desired range. We recover Urakawa's eigenvalue computation for the $SU(3)$-normal homogeneous metric on $W^{1,1}$ as a limiting case and cover all the positively curved $SU(3)\times SO(3)$-normal homogeneous realizations discovered by Wilking. By doing so, we complete Urakawa's list of the first eigenvalue on compact, simply conntected, normal homogeneous spaces with positive sectional curvature.

math.DG↗