arXiv subjects

N. C. Combe

Publications and source records attributed to N. C. Combe.

16 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.

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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.

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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.

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arXiv2026

The Koopman--von Neumann--Landau--Ginzburg theory and a Proof of the Kontsevich--Soibelman Conjecture

We show that the Hilbert space of the Koopman--von Neumann formulation of Landau--Ginzburg theory is parametrised by a real Monge--Amp\`ere domain, which carries a natural pre-Frobenius. Restricting to finite-dimensional (dually flat) exponential families, the parameter space becomes a Monge--Amp\`ere domain and a pre-Frobenius manifold. Our main theorem proves that for every Berglund--H\"ubsch--Krawitz mirror pair of Calabi--Yau orbifolds arising from an invertible polynomial, this Monge--Amp\`ere domain (the open probability simplex) is the base of a Lagrangian torus fibration on both the original and the mirror hypersurface, with dual fibres in the sense of Strominger--Yau--Zaslow. The construction recovers the SYZ picture from the Landau--Ginzburg--Koopman--von Neumann framework. In particular, this proves the Kontsevich--Soibelman conjecture (2001) for all Berglund--H\"ubsch--Krawitz mirror pairs: the base of the SYZ fibration is a Monge--Amp\`ere domain (the open simplex), and the torus fibrations on the mirror pair are dual. A toy model of cones of positive definite matrices illustrates the geometric structures.

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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-K\"ahler structure.

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arXiv2024

The Gauss--skizze decomposition is a Goresky-MacPherson stratification

We consider a new stratification of the space of configurations of $n$ marked points on the complex plane. Recall that this space can be differently interpreted as the space $^{\rm D}{\rm Pol}_{n}$ of degree $n>1$ complex, monic polynomials with distinct roots, the sum of which is 0. A stratum $A_σ$ is the set of polynomials having $P^{-1}(\mathbb{R}\cup\imath\mathbb{R})$ in the same isotopy class, relative to their asymptotic directions. We show that this stratification is a Goresky--MacPherson stratification and that from thickening strata a good cover in the sense of Čech can be constructed, allowing an explicit computation of the cohomology groups of this space.

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arXiv2022

Manin conjecture for statistical pre-Frobenius manifolds, hypercube relations and motivic Galois group in coding

This article develops, via the perspective of (arithmetic) algebraic geometry and category theory, different aspects of geometry of information. First, we describe in the terms of Eilenberg--Moore algebras over a Giry monad, the collection $Cap_n$ of all probability distributions on the measurable space $(Ω_n, \mathcal{A})$ (where $Ω$ is discrete with $n$ issues) and it turns out that there exists an embedding relation of Segre type among the product of $Cap_n$'s. We unravel hidden symmetries of these type of embeddings and show that there exists a hypercubic relation. Secondly, we show that the Manin conjecture -- initially defined concerning the diophantine geometry of Fano varieties -- is true in the case of exponential statistical manifolds, defined over a discrete sample space. Thirdly, we introduce a modified version of the parenthesised braids ($\mathbf{mPaB}$), which forms a key tool in code-correction. This modified version $\mathbf{mPaB}$ presents all types of mistakes that could occur during a transmission process. We show that the standard parenthesised braids $\mathbf{PaB}$ form a full subcategory of $\mathbf{mPaB}$. We discuss the role of the Grothendieck--Teichmüller group in relation to the modified parenthesised braids. Finally, we prove that the motivic Galois group is contained in the automorphism $Aut(\widehat{\mathbf{mPaB}}).$ We conclude by presenting an open question concerning rational points, Commutative Moufang Loops and information geometry.

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arXiv2022

Hidden symmetries of the Grothendieck--Teichmüller group

We consider the Grothendieck--Teichmüller group under a new aspect. Using real algebraic geometry and web theory we show that it preserves dihedral symmetry relations, present in the fundamental groupoids of configuration spaces of marked points on $\mathbb{C}$. The motivation of this paper is to be understood in the light of Grothendieck's initial philosophy stating that throughout hidden symmetries of the moduli spaces of curves one can shed some light on the absolute Galois group. This appears as a new development of the construction of the avatar of the Grothendieck--Teichmüller group and prepares as well the ground for studying further relations to the motivic Galois group.

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arXiv2022

Quantum SUSY operads

In a recent paper, we described a lifting of coordinate rings of groups, loops, quantum groups, etc. to the categoric setup of operads. In most examples of that paper, these rings are non--commutative. Quantum physics of the XX--th century added one more, quite nontrivial degree of freedom: coordinates might become fermionic. In their classical version, the fermionic coordinates anti--commute, and the resulting rings are called supersymmetric, or SUSY, ones. In this paper, we try to lift operads involving fermionic coordinates to quantum operads. We have to restrict ourselves by lifting operads of supersymmetric rings. We also show that $1D$ supersymmetric algebras have an operad structure, and we analyze their symmetries, through their relation to Adinkra graphs, dessins and codes.

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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.

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arXiv2021

Genus zero modular operad and absolute Galois group

In this article, we develop the geometry of canonical stratifications of the spaces $\mathcal{M}_{0,n}$ and prepare ground for studying the action of the Galois group $Gal (\overline{\mathbb{Q}} /\mathbb{Q})$ upon strata. We define and introduce a version of a gravity operad constructed for a class of moduli spaces $\mathcal{M}_{0,n}$, equipped with a hidden holomorphic involution. This additional symmetry is associated to a split quaternionic structure. We introduce a categorical framework to present this object. Interaction between the geometry, physics and the arithmetics are discussed. An important feature is that 0-divisors of the split quaternion algebra imply additional singular points, and lead to investigations concerning the geometry and mixed Hodge structures.

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arXiv2020

Cech cover of the complement of the discriminant variety. Part II: Deformations of Gauss-skizze

The configuration space of $n$ marked points on the complex plane is considered. We investigate a decomposition of this space by so-called Gauss-skizze i.e. a class of graphs being forests, introduced by Gauss. It is proved that this decomposition is a semi-algebraic topological stratification. It also forms a cell decomposition of the configuration space of $n$ marked points. Moreover, we prove that classical tools from deformation theory, ruled by a Maurer--Cartan equation, can be used only locally for Gauss-skizze. We prove that the deformation of the Gauss-skizze is governed by a Hamilton--Jacobi differential equation. This gives developments concerning Saito's Frobenius manifold. Finally, a Gauss-skizze operad is introduced. It is an enriched Fulton--MacPherson operad, topologically equivalent to the little 2-disc operad.

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arXiv2020

F-manifolds and geometry of information

The theory of $F$-manifolds, and more generally, manifolds endowed with commutative and associative multiplication of their tangent fields, was discovered and formalised in various models of quantum field theory involving algebraic and analytic geometry, at least since 1990's. The focus of this paper consists in the demonstration that various spaces of probability distributions defined and studied at least since 1960's also carry natural structures of $F$-manifolds. This fact remained somewhat hidden in various domains of the vast territory of models of information storing and transmission that are briefly surveyed here.

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arXiv2019

Symmetries of genus zero modular operad

In this article combining survey and certain research results, we introduce a categorical framework for description of symmetries of genus zero modular operad. This description merges the techniques of recent "persistence homology" studies and the classical formalism of groupoids. We show that the contravariant "poset in groupoids" embodying these symmetries, provides a new avatar of profinite Grothendieck-Teichmüller group acting upon this operad but seemingly not related with representations of the Galois group of all algebraic numbers.

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arXiv2019

Decomposition in Coxeter-chambers of the configuration space of $d$ marked points on the complex plane

Interest in Conformal Field Theories and Quantum Field Theory lead physicists to consider configuration spaces of marked points on the complex projective line, $Conf_{0,d}(\mathbb{P})$. In this paper, a real semi-algebraic stratification of $Conf_{0,d}(\mathbb{C})$, invariant under Coxeter-Weyl group is constructed, using the natural relation of this configuration space with the space $Dpol_d$ of complex monic degree $d>0$ polynomials in one variable with simple roots. This decomposition relies on subsets of $Dpol_d$ forming a good cover in the sense of Cech of $Dpol_d$ and such that each piece of the decomposition is a set of polynomials, indexed by a decorated graph reminiscent of Grothendieck's dessins d'enfant. This decomposition in Coxeter-Weyl chambers brings into light a very deep interaction between the real locus of the moduli space $\overline{\mathcal{M}}_{0,d}(\mathbb{R})$ and the complex one $\overline{\mathcal{M}}_{0,d}(\mathbb{C})$. Using this decomposition, the existence of geometric invariants of those configuration spaces has been shown. Many examples are provided. Applications of these results in braid theory are discussed, namely for the braid operad.

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arXiv2017

Connected components of real $CB_{n}$ algebraic varieties

Connected components of real algebraic varieties invariant under the $CB_{n}$-Coxeter group are investigated. In particular, we consider their maximal number and their geometric and topological properties. This provides a decomposition for the space of $CB_{n}$-algebraic varieties. We construct $CB_{n}$-polynomials using Young-posets and partitions of integers. Our results establish bounds on the number of connected components for a given set of coefficients. It turns out that this number can achieve an upper bound of $2^{n}+1$ for specific coefficients. We introduce a new method to characterize the geometry of these real algebraic varieties, using J. Cerf and A. Douady theory for varieties with angular boundary and the theory of chambers and galleries. We provide several examples that bring out the essence of these results.

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