Minerva group - Frobenius structures, Operads, Deformation Theory & Applications
Noémie Combe (Email)
+49 (0) 341 - 9959 - 775
+49 (0) 341 - 9959 - 658
Bridging algebraic geometry and probabilities & statistics
This group aims at developing and exploring new bridges between domains of algebraic geometry, algebraic topology, and information geometry. As it has been shown in , there is a deep interplay between these fundamental domains, that have until now evolved independently. Our aim is to investigate new unexpected paths, within these areas of mathematics.
In the recent work of N. Combe & Yu. Manin from 2020 (see ), there was an important discovery of a very unexpected bridge connecting algebraic geometry and the domain of probabilities and statistics, via the notion of Frobenius manifolds.
Frobenius manifolds are mathematical objects that arose in the process of axiomatization of Topological Field Theory.
Until 2020, three main classes of Frobenius manifolds were listed (and roughly presented) as:
- Quantum cohomology, in relation to Gromov--Witten invariants,
- Saito manifold (unfolding spaces of singularities), in relation to Landau--Ginzburg models,
- The moduli space of solutions to Maurer--Cartan equations appearing in the Barannikov--Kontsevich theory.
It has been proved in  that this list of Frobenius manifolds is not exhaustive and that there exists an unexpected fourth class: the class of statistical manifolds.
Statistical manifolds play a central role in geometry of information, decision theory, and machine learning. However, this object has a very rich geometry and algebraic interpretation.
Based on this viewpoint, an important task in this research program is to understand the underlying algebraic and geometric structures, which are hidden within the Frobenius manifolds, and to understand their mechanisms.
Considering the richness of the interaction of these classes of Frobenius manifolds, we also explore their ramifications to other domains of mathematics, such as the Grothendieck—Teichmüller theory (see ).
- Leipzig seminar on Algebra, Algebraic Geometry and Algebraic Topology
(Selected Thursdays 15:15 - 16:45, 2021, Zoom Seminar)
My research began at the University of Geneva with Stanislav K. Smirnov, when I was a third year bachelor student, in 2012. Two years after he won his Fields medal he gave a lecture with no prerequisites about his research topic, targeted at a large audience (from the large range of Bachelor students to Professors). There was an exam test for the students and I was given a project concerning the Cardy--Smirnov formula, a beautiful subject mixing probabilities, Schramm--Lowner process (curves given by differential equations called SLE_kappa) and Hausdorff dimension. Working on this subject, stimulated my research interests and contributions.
Later on, I drifted to a more algebraic geometry/topological aspect of mathematics: moduli spaces. I contributed to the problem of counting the number of connected components of real algebraic varieties of a given degree (which is a problem deeply related to the Hilbert 16th problem).This made my Master thesis, under the supervision of Daniel Coray.
Immediately, after my Master thesis, I was awarded a prestigious Labex grant to become a research assistant and PhD student at the University of Aix-Marseille and Sorbonne University, Paris 6. My PhD was supervised by Prof. Dr. Bernard Coupet and Prof. Dr. Norbert A'Campo on configuration spaces.
After receiving my PhD certificate in 2018, I moved to Max Planck Institute Bonn (Germany) for one year and a half to work as a postdoctoral fellow in moduli spaces and operads, with Prof. Dr. Yu. I. Manin. In February 2020, I was appointed as a Minerva fellow of the Max Planck Society, and chose to move and work in Max Planck Institute for Maths in Leipzig as a Minerva group leader.