Periodic Reporting for period 4 - CombiTop (New Interactions of Combinatorics through Topological Expansions, at the crossroads of Probability, Graph theory, and Mathematical Physics)
Periodo di rendicontazione: 2021-09-01 al 2022-08-31
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.
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.
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.