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

TL;DR

This paper explores the intersection of algebraic geometry, category theory, and information geometry, demonstrating new geometric structures, validating the Manin conjecture for statistical manifolds, and linking Galois groups to coding theory.

Contribution

It introduces a hypercubic relation in probability distributions, proves the Manin conjecture for exponential statistical manifolds, and connects motivic Galois groups with automorphisms of modified parenthesised braids.

Findings

Existence of hypercubic relations among probability distributions.

Validation of the Manin conjecture for exponential statistical manifolds.

Motivic Galois group is contained in automorphisms of modified parenthesised braids.

Abstract

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 $(\Omega_n, \mathcal{A})$ (where $\Omega$ 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--Teichm\"uller 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.