arXiv subjects

H. K. Nencka

Publications and source records attributed to H. K. Nencka.

5 recordsLinked to original sources

arXiv2026

Shape Theory $\&$ TDA via the Atiyah--Molino Reconstruction

Reconstruction problems lie at the very heart of both mathematics and science, posing the fundamental challenge: \emph{How does one reconstruct a hidden structure from incomplete, fragmented, or distorted data?} In this paper, we introduce a new approach that harnesses the insights of the Vaisman Atiyah--Molino framework. In contrast to conventional methods that depend on persistent homology, our approach exploits the concept of the Vaisman centroid---an intrinsic invariant that encapsulates the averaged geometry of a data set---to resolve the inherent ambiguities of inverse problems. In the present paper, we focus on the theory and applications of the Vaisman centroid, offering a new perspective for Topological Data Analysis that eschews persistent homology in favour of a unified geometric paradigm. A subsequent paper will extend these ideas to a complete reconstruction scheme via the Atiyah--Molino framework. Our method provides a robust and computationally tractable framework for the recovery of hidden structures while opening new avenues for the analysis of high-dimensional and noisy data across the mathematical sciences.

math.DG↗

arXiv2026

Image Recognition via Vaisman--Neifeld's Geometry

We introduce a new approach to the reconstruction of hidden structures from incomplete data, unifying techniques from geometric integration and topological analysis within the frameworks of Vaisman and Neifeld. Our method employs a refined geometric decomposition of configuration spaces into invariant foliations and moment maps, thereby addressing the intrinsic ambiguities of underdetermined inverse problems. By combining Vaisman's insights into symmetry with Neifeld's analytical methodologies, we establish a robust, noise-resistant framework that ensures computational tractability while providing a unified perspective on reconstruction in imaging and structural analysis. This approach enables applications across diverse scientific domains and highlights the interplay between geometry and topology in the solution of inverse problems.

math.AG↗

arXiv2026

Foliated Geometry of Inverse Problems: Torsion, Curvature Duality, and Near-Associativity

We present a geometric framework for reconstruction problems based on Vaisman foliations and Atiyah--Molino sequences. Independent projections induce transverse foliations and dual connections; vanishing torsion and curvature duality guarantee unique, path-independent reconstruction, while obstructions yield non-associative quasigroupoids. Toric symmetry provides equivariant uniqueness. Applications to generative AI imputation and cryo-electron microscopy demonstrate the framework's practical power, unifying differential geometry with data-driven inverse problems.

math.DG↗

arXiv2026

On Duality, Legendre Bundles and Deformations

We introduce the Legendre bundle, a geometric structure encoding the essential duality of dually flat (Hessian) manifolds, and demonstrate that both exponential families in information geometry and a natural class of quantum field theories -- which we term Hessian QFTs -- arise as distinct realisations of this single framework. The Legendre bundle is shown to carry a canonical para-Kähler structure.

math.DG↗

arXiv2021

Algebraic properties of the information geometry's fourth Frobenius manifold

Recently, it has been shown that the statistical manifold, related to exponential families, has a Frobenius manifold structure and appears as the fourth class of Frobenius manifolds. It has a structure of a projective manifold over a rank two Frobenius algebra $\frak{A}$, being the algebra of paracomplex numbers and generated by $1, ε$ such that $ε^2=1$. This result is a key step towards an algebraization of the results concerning the manifold of probability distributions and thus offers a new perspective on it. In this paper, we prove that the fourth Frobenius manifold is decomposed into a pair of symmetric totally geodesic pseudo-Riemannian submanifolds, each of which correspond to a module over an ideal of $\frak{A}$. This pair of ideals are othogonal idempotents. The symmetry is obtained under the Peirce mirror.

math.AG↗