arXiv subjects

Noémie C. Combe

Publications and source records attributed to Noémie C. Combe.

10 recordsLinked to original sources

arXiv2026

Wishart Matrices and Quantum Geometry: Foundations and Applications in Quantum Information

We present a unified framework for the study of Wishart matrices (W_p(n,Σ)), which generalize the chi-squared distribution to matrix-variate settings and model the covariance structure of multivariate Gaussian data. After recalling their defining properties - additivity under independent summation (W_1 + W_2 \sim W_p(n_1+n_2,Σ)), equivariance under linear maps (A W A^T \sim W_q(n,AΣA^T)), and their role as sample covariance matrices - we embed the positive-definite cone (S_p^+) within Monge-Ampere geometry. Here (S_p^+) acquires a Hessian manifold structure with affine-invariant metric and volume form (ω= \det(Σ)^{-(p+1)/2},dΣ), under which the Wishart density acts as a soliton of natural geometric flows. We then show that the collection of Wishart distributions forms a symmetric monoidal category (\mathcal{W}), whose objects are (W_p(n,Σ)) and whose morphisms are linear maps (A:\mathbb{R}^p\to\mathbb{R}^q). The tensor product encodes block-diagonal coupling, with braiding given by block permutation, and the axioms enforce Monge-Ampere functoriality, additivity, and convex duality via the Legendre transform. Applications to quantum error correction are discussed: Wishart laws model correlated noise, Wasserstein geodesics optimize error-mitigation cost, tensor structure captures independent error channels, and Legendre duality underpins entropy-driven decoding.

math.DG↗

arXiv2026

The Infinite Sphere and Galois Belyi maps

We show that the space of Belyi maps admits a natural parametrization by an infinite-dimensional sphere arising from Voiculescu's theory of noncommutative probability spaces. We show that this sphere decomposes into sectors, each of which corresponds to a class of Belyi maps distinguished up to isomorphism by their monodromy, encoded by a finite-index subgroup of F2. For Galois Belyi maps, our correspondence between spectral sectors of the infinite sphere and algebraic quotients of F2 yields a genuine bijection. Within this framework, distinct sectors of the sphere capture the algebraic constraints imposed on the monodromy, thereby providing a geometric organization of Belyi maps according to their associated group-theoretic data.

math.AG↗

arXiv2025

Proving the Grothendieck--Teichmüller Conjecture for Profinite Spaces & The Galois Grothendieck Path Integral

We establish that the Grothendieck-Teichmuller conjecture, which predicts an isomorphism between the Grothendieck-Teichmuller group GT and the absolute Galois group of rational numbers Gal, holds in the setting of profinite spaces. To access arithmetic information within this framework, we introduce a generalization of the notion of path integrals, defined simultaneously for GT and Gal. This construction reveals new arithmetic invariants for both groups, capturing, in particular, the rationality of periods associated with GT and the rational structure of the coefficients in the Drinfeld associator. Moreover, this perspective provides a mechanism to detect the descent of cohomology classes to Q under the action of Gal, and more broadly, a method to compute ranks of arithmetic objects. Furthermore, we introduce an algorithm, which we call the Cubic Matrioshka, encoding paths in both GT and Gal as unique infinite binary sequences. The compatibility of these binary encodings reflects the underlying homeomorphism and ensures that symmetries of GT are faithfully mirrored in the corresponding binary representation within Gal.

math.AG↗

arXiv2025

Shape Theory via the Atiyah--Molino Reconstruction and Deformations

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.

math.DG↗

arXiv2025

Quantum Geometry insights in Deep Learning

In this paper, we explore the fundamental role of the Monge-Ampère equation in deep learning, particularly in the context of Boltzmann machines and energy-based models. We first review the structure of Boltzmann learning and its relation to free energy minimization. We then establish a connection between optimal transport theory and deep learning, demonstrating how the Monge-Ampère equation governs probability transformations in generative models. Additionally, we provide insights from quantum geometry, showing that the space of covariance matrices arising in the learning process coincides with the Connes-Araki-Haagerup (CAH) cone in von Neumann algebra theory. Furthermore, we introduce an alternative approach based on renormalization group (RG) flow, which, while distinct from the optimal transport perspective, reveals another manifestation of the Monge-Ampère domain in learning dynamics. This dual perspective offers a deeper mathematical understanding of hierarchical feature learning, bridging concepts from statistical mechanics, quantum geometry, and deep learning theory.

cs.LG↗

arXiv2025

Exploring information geometry: Recent Advances and Connections to Topological Field Theory

This introductory text arises from a lecture given in Göteborg, Sweden, given by the first author and is intended for undergraduate students, as well as for any mathematically inclined reader wishing to explore a synthesis of ideas connecting geometry and statistics. At its core, this work seeks to illustrate the profound and yet natural interplay between differential geometry, probability theory, and the rich algebraic structures encoded in (pre-)Frobenius manifolds. The exposition is structured into three principal parts. The first part provides a concise introduction to differential topology and geometry, emphasizing the role of smooth manifolds, connections, and curvature in the formulation of geometric structures. The second part is devoted to probability, measures, and statistics, where the notion of a probability space is refined into a geometric object, thus paving the way for a deeper mathematical understanding of statistical models. Finally, in the third part, we introduce (pre-)Frobenius manifolds, revealing their surprising connection to exponential families of probability distributions and, discuss more broadly, their role in the geometry of information. At the end of those three parts the reader will find stimulating exercises. By bringing together these seemingly distant disciplines, we aim to highlight the natural emergence of geometric structures in statistical theory. This work does not seek to be exhaustive but rather to provide the reader with a pathway into a domain of mathematics that is still in its formative stages, where many fundamental questions remain open. The text is accessible without requiring advanced prerequisites and should serve as an invitation to further exploration.

math.DG↗

arXiv2025

On Multiquantum Bits, Segre Embeddings and Coxeter Chambers

This work explores the interplay between quantum information theory, algebraic geometry, and number theory, with a particular focus on multiqubit systems, their entanglement structure, and their classification via geometric embeddings. The Segre embedding, a fundamental construction in algebraic geometry, provides an algebraic framework to distinguish separable and entangled states, encoding quantum correlations in projective geometry. We develop a systematic study of qubit moduli spaces, illustrating the geometric structure of entanglement through hypercube constructions and Coxeter chamber decompositions. We establish a bijection between the Segre embeddings of tensor products of projective spaces and binary words of length $n-1$, structured as an $(n-1)$-dimensional hypercube, where adjacency corresponds to a single Segre operation. This reveals a combinatorial structure underlying the hierarchy of embeddings, with direct implications for quantum error correction schemes. The symmetry of the Segre variety under the Coxeter group of type $A$ allows us to analyze quantum states and errors through the lens of reflection groups, viewing separable states as lying in distinct Coxeter chambers on a Segre variety. The transitive action of the permutation group on these chambers provides a natural method for tracking errors in quantum states and potentially reversing them. Beyond foundational aspects, we highlight relations between Segre varieties and Dixon elliptic curves, drawing connections between entanglement and number theory.

quant-ph↗

arXiv2025

Landau-Ginzburg models, Monge-Ampère domains and (pre-)Frobenius manifolds

Kontsevich suggested that the Landau-Ginzburg model presents a good formalism for homological mirror symmetry. In this paper we propose to investigate the LG theory from the viewpoint of Koopman-von Neumann's construction. New advances are thus provided, namely regarding a conjecture of Kontsevich-Soibelman (on a version of the Strominger-Yau-Zaslow mirror problem). We show that there exists a Monge-Ampère domain Y, generated by a space of probability densities parametrising mirror dual Calabi-Yau manifolds. This provides torus fibrations over Y. The mirror pairs are obtained via the Berglund-Hubsch-Krawitz construction. We also show that the Monge-Ampère manifolds are pre-Frobenius manifolds. Our method allows to recover certain results concerning Lagrangian torus fibrations. We illustrate our construction on a concrete toy model, which allows us, additionally to deduce a relation between von Neumann algebras, Monge-Ampère manifolds and pre-Frobenius manifolds.

math.AG↗

arXiv2025

Maximum Likelihood, permutohedra and Associativity Equations

We consider the cone of concentration matrices related to linear concentration models and Wishart laws. We prove that this cone is a Monge--Ampère domain and that the log-likelihood function generates its potential function at the identity. The tangent sheaf carries the structure of a pre-Lie algebra. We also show that the moduli space of diagonal matrices parameterizing the polyhedral spectrahedron satisfies the Associativity Equations, a notion central in mirror symmetry, and that its compactification is a toric variety associated to a permutohedron, reminiscent to Losev--Manin spaces. Finally we introduce Frobenius residuals: these are connected components of the compactified Frobenius manifold of diagonal matrices, generated by the Białynicki--Birula cells. We prove that the Maximum Likelihood degree is indexed by components lying on those Frobenius residuals.

math.AG↗

arXiv2024

Learning on hexagonal structures and Monge-Ampère operators

Dually flat statistical manifolds provide a rich toolbox for investigations around the learning process. We prove that such manifolds are Monge-Ampère manifolds. Examples of such manifolds include the space of exponential probability distributions on finite sets and the Boltzmann manifolds. Our investigations of Boltzmann manifolds lead us to prove that Monge-Ampère operators control learning methods for Boltzmann machines. Using local trivial fibrations (webs) we demonstrate that on such manifolds the webs are parallelizable and can be constructed using a generalisation of Ceva's theorem. Assuming that our domain satisfies certain axioms of 2D topological quantum field theory we show that locally the learning can be defined on hexagonal structures. This brings a new geometric perspective for defining the optimal learning process.

math.DG↗