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By a combinatorial map we mean a graph embedded in the two dimensional surface.
Enumeration of combinatorial maps and fully simple maps (maps with some additional
conditions on them) are governed by Chekhov-Eynard-Orantin topological recursion --- a
universal recursive procedure which, by the small amount of the initial data (Riemann
surface with two functions x and y on it), produces symmetric meromorphic
n-differentials possessing all the information about the underlying enumerative
problem.
The duality between maps and fully simple maps goes through the monotone Hurwitz
numbers was obtained by G.Borot, S.Charbonnier, N.Do and E.Garcia-Failde in 2019. In
the talk I will explain this result and, using the combinatorics of the symmetric
group and Fock space formalism, will describe its connection to the general x-y
duality in topological recursion.
The talk is based on the series of papers joined with A. Alexandrov,
P.Dunin-Barkowski, M.Kazarian and S.Shadrin https://arxiv.org/abs/2106.08368,
https://arxiv.org/abs/2212.00320
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