Preprints:

Volumes of moduli spaces of directed ribbon graphs and Cut and Join operators arxiv

In this paper, we investigate the algebraic structure underlying the acyclic decomposition. This decomposition applies to directed metric ribbon graphs and enables the recursive computation of the volumes of their moduli spaces. Building on this, we define integral operators with these volumes and show that they satisfy a Cut-and-Join type equation. Furthermore, we demonstrate that a suitable specialization of these operators gives rise to a generating series for dessins d'enfants.

Cutting orientable ribbon graphs arxiv

We study oriented metric ribbon graphs. We show that it's possible to decompose these graphs in some canonical way by performing surgery along appropriate multi-curves. This result provides a recursion scheme for the volumes of the moduli space of 4−valent metric ribbon graphs, which can be interpreted as an oriented version of the topological recursion. We give applications to counting dessins d'enfants in a particular case.

Length spectrum of large genus random metric maps with Alessandro Giacchetto and Mingkun Liu arxiv

We study the length of short cycles on metric ribbon graphs of large genus using a Teichmüller theory approach. We establish that, as the genus tends to infinity, the length spectrum converges to a Poisson point process with an explicit intensity.

Average diffusion rate of Ehrenfest Wind-tree billiards arxiv

I study the diffusion in the Erenfest wind tree billiard, using tool from dynamics over moduli space of flat surfaces

Phd thesis:

Geometric recursion and volumes of moduli spaces : Oriented ribbon graphs, acyclic decomposion, “Cut-and-Join” operators Hall

In this thesis we study the relations between topological and geometric recursions and Masur Veech volumes of moduli spaces of quadratic and Abelian differentials. We chose to study ribbon graphs because they can be used to compute these volumes. In the case of trivalent ribbon graphs we give a geometric recursion formula that was also independently found in "On the Kontsevich geometry of the combinatorial Teichmüller space". We also study oriented ribbon graphs, in this case we found decomposition of graphs that we call "The acyclic decomposition''. This decomposition allow to decompose general oriented ribbon graphs into graphs with only one vertex. Using this we are able to compute volumes of their moduli spaces. We relate the acyclic decomposition to cut and joins operators. At the end of the memoir we study degenerations of ribbon graphs and show that volumes of moduli spaces of oriented ribbon graphs admit continuous extensions.

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