This project, originates in the beautiful interaction between algebraic geometry, algebraic topology and certain aspects of mathematical physics.
This synergy gave birth to some of the most spectacular developments of the last 50 decades in pure mathematics. This has also left deep open problems, which present a stimulating source of inspiration.
One result of this interaction was the creation of the theory of quantum cohomology, its connections to the world of birational geometry and the subsequent discovery of its connections with one of the central objects of number theory: the Galois group of the field of all algebraic numbers. Frobenius algebras and Frobenius manifolds, also emerged from this synergy and present different facets of one big problem.
Open problems where it is question of Grothendieck—Teichmuller groups, Multiple Zeta Values, configuration spaces, duality patterns and intersection of birational cycles are a motivation for this research proposal.
I acknowledge support from the GRANT POLONEZ BIS 3.
This research is part of the project No. 2022/47/P/ST1/01177 co-founded by the National Service Centre and the European Union's Horizon 2020 research and innovation program, under the Marie Sklodowska Curie grant agreement No. 945339.
Publications
N. C. Combe, Proving the Grothendieck-Teichmüller Conjecture for Profinite Spaces & The Galois Grothendieck Path Integral. ArXiv:2503.1306
N. C. Combe, Quantum Geometry insights in Deep Learning, arXiv:2503.02655
N. C. Combe, On Multiquantum Bits, Segre Embeddings and Coxeter Chambers, http://arxiv.org/abs/2502.00461
N. C. Combe, Maximum Likelihood, permutohedra and Associativity Equations. arXiv:2501.01345
N. C. Combe, Wishart cones and quantum geometry, ArXiv[2412.12289] Accepted for publication!
N.C. Combe, On the geometry of Kahler--Frobenius manifolds and their classification, ArXiv[2411.14362]
N. C. Combe, Learning on hexagonal structures and Monge--Ampere operators ArXiv[2412.04407]
Goals of the F-SODA research group (Minerva grant)
The goal of this group is to develop and establish new mathematical bridges between recent and old developments on Frobenius manifolds.
Frobenius manifolds are mathematical objects that arose in the process of axiomatisation of Quantum 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 and the moduli space of solutions to Maurer--Cartan equations appearing in the Barannikov--Kontsevich theory.
In 2020, in a work of N. Combe & Yu. Manin, there was the great discovery of a very unexpected bridge between algebraic geometry and the domain of probabilities and statistics, through the notion of Frobenius manifolds.
It has been proved that this list of Frobenius manifolds is not exhaustive and that there exists a fourth class, which no one expected: the class of statistical manifolds.
A statistical manifold is a space of probability distributions. It plays a central role in geometry of information, decision theory and machine learning. However, this object has a very rich geometry and algebraic interpretation.
An important task in this research program is to understand the underlying algebraic and geometric structures, which are hidden within the Frobenius manifolds and understand their relations and ramifications to each other and to other domains of mathematics.
Noémie C. Combe, Yuri I. Manin, F-manifolds and geometry of information, Bull. London Maths Soc. Volume 52, Issue 5. (2020)
Noémie C. Combe, Yuri I. Manin, Symmetries of genus zero modular operads , Integrability, Quantization, and Geometry - 'Dubrovin memorial volume' Proceedings Symposia in Pure Mathematics, American Mathematical Society (2020)
RESEARCH FELLOWSHIP (2018-2019)
Max-Planck Institute for Mathematics in Bonn (DE).
MENTOR: Yuri I. Manin.
The interplay between Grothendieck–Teichmüller theory, monodromy phenomena, and the Kashiwara–Vergne conjecture forms one of the deepest crossroads in modern mathematics, where arithmetic geometry, low-dimensional topology, and Lie-theoretic representation theory converge. A research program centered on this triad offers the possibility of unraveling the hidden symmetry structures governing fundamental groups of algebraic varieties, deformation quantization, and universal associators. Grothendieck’s vision of dessins d’enfants and the Teichmüller tower provides a conceptual scaffolding for understanding the absolute Galois group through braid-like symmetries, while monodromy problems translate geometric and analytic continuity into powerful algebraic constraints. The Kashiwara–Vergne conjecture, meanwhile, connects deep harmonic analysis on Lie groups with Drinfeld associators, suggesting that the keys to these structures may lie in explicitly constructing and comparing deformation-theoretic symmetries. A coherent program exploring these themes has the potential not only to resolve long-standing conjectures but to reveal a unifying framework
Noémie C. Combe, Yuri I. Manin, Genus zero modular operad and absolute Galois group, Arxiv: 1907.10313 North-W. Eur. J. of Math., Vol 8, Pages 25-60, (2022)
Noémie C. Combe, Yuri I. Manin, Symmetries of genus zero modular operads , Integrability, Quantization, and Geometry - 'Dubrovin memorial volume' Proceedings Symposia in Pure Mathematics, American Mathematical Society (2020)