arXiv subjects

Noemie C. Combe

Publications and source records attributed to Noemie C. Combe.

5 recordsLinked to original sources

arXiv2024

Wishart cones and quantum geometry

An important object appearing in the framework of the Tomita--Takesaki theory is an invariant cone under the modular automorphism group of von Neumann algebras. As a result of the connection between von Neumann algebras and quantum field theory, von Neumann algebras have become increasingly important for (higher) category theory and topology. We show explicitly how an example of a class of cones discovered by Connes--Araki--Haagerup (CAH), invariant under the modular automorphism group, are related to Wishart laws and information geometry. Given its relation to 2D quantum field theory this highlights new relations between (quantum) information geometry and quantum geometry.

math.OA↗

arXiv2023

On Frobenius structures in symmetric cones

We prove that in any strictly convex symmetric cone $Ω$ there exists a non empty locus where the WDVV equation is satisfied (i.e. there exists a hyperplane being a Frobenius manifold). This result holds over any real division algebra (with a restriction to the rank 3 case if we consider the field $\mathbb{O}$) but also on their linear combinations. This theorem holds as well in the case of pseudo-Riemannian geometry, in particular for a Lorentz symmetric cone of Anti-de-Sitter type. Our statement can be considered as a generalisation of a result by Ferapontov--Kruglikov--Novikov and Mokhov. Our construction is achieved by merging two different approaches: an algebraic/geometric one and the analytic approach given by Calabi in his investigations on the Monge--Ampère equation for the case of affine hyperspheres.

math.AG↗

arXiv2022

A note on exponential varieties, statistical manifolds and Frobenius structures

New relations between algebraic geometry, information theory and Topological Field Theory are developed. One considers models of databases subject to noise i.e. probability distributions on finite sets, related to exponential families. We prove explicitly that these manifolds have the structure of a pre-Frobenius manifold, being a pre-structure appearing in the process of axiomatisation of Topological Field Theory. On one hand, this allows us to develop relations to algebraic geometry, by proving explicitly that a statistical pre-Frobenius manifold forms an algebraic variety over $\mathbb{Q}$ (i.e. $\mathbb{Q}$-toric variety). On the other hand, this allows further developments of recent results concerning the hidden symmetries of those objects. Using classical web theory, it has been shown that those symmetries have the structure of Commutative Moufang Loops. Our result allows to develop more algebraically this statement, in a two-fold way. First, from an algebraic point of view it follows that statistical pre-Frobenius manifolds are equipped with algebraizable webs. Secondly, from the differential geometry point of view, it follows that these webs are hexagonal and isoclinic. This statement is important since it directly impacts the geometric properties of the {\it statistical data}, which are tightly related to the webs. Hence, this allows deeper connections to the branch of algebraic statistics, which is concerned with the development of techniques in algebraic geometry, commutative algebra, to address problems in statistics and its applications. Examples are provided and discussed.

math.AG↗

arXiv2020

Birational maps and Nori motives

In this note, we sketch an approach to the problems of equivariant birational geometry developed by M. Kontsevich and Yu. Tschinkel, where Burnside invariants were introduced. We are making explicit the role of Nori constructions in this environment.

math.AG↗

arXiv2020

Dessins for Modular Operad and Grothendieck-Teichmuller Group

A part of Grothendieck's program for studying the Galois group $G_{\mathbb Q}$ of the field of all algebraic numbers $\overline{\mathbb Q}$ emerged from his insight that one should lift its action upon $\overline{\mathbb Q}$ to the action of $G_{\mathbb Q}$ upon the (appropriately defined) profinite completion of $π_1({\mathbb P}^1 \setminus \{0,1, \infty\})$. The latter admits a good combinatorial encoding via finite graphs "dessins d'enfant". This part was actively developing during the last decades, starting with foundational works of A. Belyi, V. Drinfeld and Y. Ihara. Our brief note concerns another part of Grothendieck program, in which its geometric environment is extended to moduli spaces of algebraic curves, more specifically, stable curves of genus zero with marked/labelled points. Our main goal is to show that dual graphs of such curves may play the role of "modular dessins" in an appropriate operadic context.

math.AG↗