New Interactions of Combinatorics through Topological Expansions, at the crossroads of Probability, Graph theory, and Mathematical Physics

The CombiTop project is situated at the intersection between several areas of mathematics, such as combinatorics, probability, and mathematical physics. It is centered about certain combinatorial objects called maps, which describe the embedding (the drawing) of a graph, made by vertices and connection between them, on a surface such as a sphere, a torus, a double-torus, etc. Maps are ubiquitous in all sciences and they play a deep role in geometry, in probability theory, in mathematical physics, either because they serve as discrete models for deterministic or random surfaces appearing in these fields, or because they index some of the mathematical objects that appear at the abstract level in these theories.
All these approaches have contributed to make the theory of maps very active, in particular at the enumerative level (whose main question is to try to find formulas to count maps of a given type). In particular, recent progresses on the most combinatorial, concrete, approaches, have led to breakthroughs in probability theory by enabling the construction of Brownian maps, a universal mathematical object describing the behaviour of a random continuum surface. The goal of the CombiTop project is to look for a further unification of all these approaches, taking as a starting point the purely combinatorial level. By confronting this viewpoint with other fields, we expect to unveil new tools, new methods, new results, but also new directions and new questions for each of them.

A key achievement of the project is the proof by our then PhD student Baptiste Louf and Thomas Budzinski (not a project member) of the Benjamini-Curien conjecture, which describes in a very precise way the behaviour of maps whose number of edges grows rapidly at the same rate as the number of handles of the surface they are drawn on. This result relies on a number of innovations at the technical level, coming from both probability and combinatorics.

The next important result is the proof by the PI Guillaume Chapuy, with coauthors Alexander Alexandrov, Bertrand Eynard, John Harnad (not project members), that deep structures originating in enumerative geometry and called "Weighted Hurwitz numbers" satisfy a set of formulas known as the Topological Recursion and originating in String Theory in mathematical physics. This puts several results obtained by several authors in these fields under a common framework, via again the introduction of novel technical methods, inspired by mixing the combinatorial, analytic, and operatorial viewpoints. This work has been completed with the proof by Guillaume Chapuy with Valentin Bonzom (not project member), Séverin Charbonnier and Elba Garcia-Failde (two postdocs on the project) that the topological recursion works in the presence of "internal faces", which gives an infinite parameter generalization of Eynard's famous solution of the 2-Matrix Model.

In another direction, the joint paper in 2020 by Guillaume Chapuy with Maciej Dołęga (not project member) about branched covered and Jack expansions was an unexpected breakthrough in the theory of Jack polynomials and their link with map enumeration. At its very core, it relies on a new "operator" approach to the link between maps and symmetric functions. A set of results by these two authors with Valentin Bonzom (not project member) method develop further the first steps of a theory of "b-deformed" Hurwitz numbers.

Finally, Guillaume Chapuy and Theo Douvropoulos (postdoc on the project) made crucial steps in the unification of map enumeration and Coxeter combinatorics, including a version of the Matrix-Tree theorem for reflection groups, and the first case-free proof of the Deligne-Loojienga formula counting Coxeter factorisations in reflection groups.

All these results were submitted to peer-reviewed international journals, and presented in international seminars or conferences.

A key achievement of the project is the proof by our then PhD student Baptiste Louf and Thomas Budzinski (not a project member) of the Benjamini-Curien conjecture, which describes in a very precise way the behaviour of maps whose number of edges grows rapidly at the same rate as the number of handles of the surface they are drawn on. This result relies on a number of innovations at the technical level, coming from both probability and combinatorics.

The next important result is the proof by the PI Guillaume Chapuy, with coauthors Alexander Alexandrov, Bertrand Eynard, John Harnad (not project members), that deep structures originating in enumerative geometry and called "Weighted Hurwitz numbers" satisfy a set of formulas known as the Topological Recursion and originating in String Theory in mathematical physics. This puts several results obtained by several authors in these fields under a common framework, via again the introduction of novel technical methods, inspired by mixing the combinatorial, analytic, and operatorial viewpoints. This work has been completed with the proof by Guillaume Chapuy with Valentin Bonzom (not project member), Séverin Charbonnier and Elba Garcia-Failde (two postdocs on the project) that the topological recursion works in the presence of "internal faces", which gives an infinite parameter generalization of Eynard's famous solution of the 2-Matrix Model.

In another direction, the joint paper in 2020 by Guillaume Chapuy with Maciej Dołęga (not project member) about branched covered and Jack expansions was an unexpected breakthrough in the theory of Jack polynomials and their link with map enumeration. At its very core, it relies on a new "operator" approach to the link between maps and symmetric functions. A set of results by these two authors with Valentin Bonzom (not project member) method develop further the first steps of a theory of "b-deformed" Hurwitz numbers.

Finally, Guillaume Chapuy and Theo Douvropoulos (postdoc on the project) made crucial steps in the unification of map enumeration and Coxeter combinatorics, including a version of the Matrix-Tree theorem for reflection groups, and the first case-free proof of the Deligne-Loojienga formula counting Coxeter factorisations in reflection groups.

Periodic Reporting for period 4 - CombiTop (New Interactions of Combinatorics through Topological Expansions, at the crossroads of Probability, Graph theory, and Mathematical Physics)

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