12 papers
On Duality, Legendre Bundles and Deformations
N. C. Combe, P. G. Combe, H. K. Nencka
We introduce the Legendre bundle, a geometric structure encoding the essential duality of dually flat (Hessian) manifolds, and demonstrate that both exponential families in informa…
The Infinite Sphere and Galois Belyi maps
Noémie C. Combe
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 sh…
Proving the Grothendieck--Teichmüller Conjecture for Profinite Spaces & The Galois Grothendieck Path Integral
Noémie C. Combe
We establish that the Grothendieck-Teichmuller conjecture, which predicts an isomorphism between the Grothendieck-Teichmuller group GT and the absolute Galois group of rational num…
Shape Theory via the Atiyah--Molino Reconstruction and Deformations
Noémie C. Combe, Hanna N. Nencka
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 in…
Quantum Geometry insights in Deep Learning
Noémie C. Combe
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 r…
Exploring information geometry: Recent Advances and Connections to Topological Field Theory
Noémie C. Combe, Philippe G. Combe, Hanna K. Nencka
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 inc…