## Periodic Reporting for period 2 - CombiTop (New Interactions of Combinatorics through Topological Expansions, at the crossroads of Probability, Graph theory, and Mathematical Physics)

Reporting period: 2018-09-01 to 2020-02-29

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.

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.

"We obtained a first set results going in the direction of understanding better the connection between the ""Fermionic"" viewpoint on map enumeration, first developed by mathematical physicists, and the combinatorial viewpoint. With these methods, our PhD student Baptiste Louf obtained the first recurrence formulas that enable to compute, in practice, the number of inequivalent graph drawings with given number of edges and even degrees, on a surface with a given number of handles.

Our most important result so far is the proof by Baptiste Louf and Thomas Budzinski (not a project member) of the Benjamini-Curien conjecture, which had resisted all attempts from experts for several years. Their result 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.

Our 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.

These key results take place within a more general effort to jointly advance the theory of map enumeration from the algebraic, combinatorial, and probabilistic viewpoint, and we have obtained a coherent set of results going in this direction."

Our most important result so far is the proof by Baptiste Louf and Thomas Budzinski (not a project member) of the Benjamini-Curien conjecture, which had resisted all attempts from experts for several years. Their result 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.

Our 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.

These key results take place within a more general effort to jointly advance the theory of map enumeration from the algebraic, combinatorial, and probabilistic viewpoint, and we have obtained a coherent set of results going in this direction."

The main results described above are significant progresses beyond the state of the art. For the remainder of the project, several of the original questions are still open, including the construction of general bijections for Fermionic formulas, the behaviour of Brownian maps of genus g via their Fermionic observables, or the development of the algebro-combinatorial theory of map enumeration.