Shape Theory via the Atiyah--Molino Reconstruction and Deformations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908297233170432 |
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| author | Combe, Noémie C. Nencka, Hanna N. |
| author_facet | Combe, Noémie C. Nencka, Hanna N. |
| contents | Reconstruction problems lie at the very heart of both mathematics and science, posing the enigmatic challenge: \emph{How does one resurrect a hidden structure from the shards of incomplete, fragmented, or distorted data?} In this paper, we introduce a new approach that harnesses the profound insights of the Vaisman Atiyah--Molino framework. Our method renders the reconstruction problem computationally tractable while exhibiting exceptional robustness in the presence of noise. Central to our theory is the Hantjies tensor -- a curvature-like invariant that precisely quantifies noise propagation and enables error-bounded reconstructions. This synthesis of differential geometry, integral analysis, and algebraic topology not only resolves long-standing ambiguities in inverse problems but also paves the way for transformative applications across a broad spectrum of scientific disciplines. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01831 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shape Theory via the Atiyah--Molino Reconstruction and Deformations Combe, Noémie C. Nencka, Hanna N. Differential Geometry Reconstruction problems lie at the very heart of both mathematics and science, posing the enigmatic challenge: \emph{How does one resurrect a hidden structure from the shards of incomplete, fragmented, or distorted data?} In this paper, we introduce a new approach that harnesses the profound insights of the Vaisman Atiyah--Molino framework. Our method renders the reconstruction problem computationally tractable while exhibiting exceptional robustness in the presence of noise. Central to our theory is the Hantjies tensor -- a curvature-like invariant that precisely quantifies noise propagation and enables error-bounded reconstructions. This synthesis of differential geometry, integral analysis, and algebraic topology not only resolves long-standing ambiguities in inverse problems but also paves the way for transformative applications across a broad spectrum of scientific disciplines. |
| title | Shape Theory via the Atiyah--Molino Reconstruction and Deformations |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2504.01831 |