Some structural and enumerative aspects of discrete surfaces and PL-manifolds

TL;DR

This paper explores algebraic and enumerative properties of discrete surfaces and PL-manifolds, focusing on generalizations of bipartite maps, their universal structures, and higher-dimensional triangulations, revealing new combinatorial and topological insights.

Contribution

It introduces new connections between map generalizations and universal structures like topological recursion and hierarchies, extending these concepts to higher dimensions and complex combinatorial models.

Findings

Topological recursion applies to oriented, double, weighted Hurwitz numbers.

Unoriented weighted Hurwitz numbers can transition from KP to BKP hierarchy.

Certain 3D gluings are in bijection with trees, indicating universality.

Abstract

This manuscript recounts some of the author's contributions to algebraic and enumerative combinatorics. We have focused on two types of generalizations of bipartite maps, which are bipartite graphs embedded on surfaces. Maps are known to appear in many areas of theoretical physics and discrete mathematics, but one key interest for fundamental computer science is how multi-facetted they are in the sense that multiple encodings exist which are not interchangeable, like Tutte's/loop equations, the topological recursion, the KP hierarchy and numerous bijections which made the field so rich. One generalization we considered is weighted Hurwitz numbers, including constellations, monotone Hurwitz numbers and the unoriented versions of Chapuy-Do\l\k{e}ga. We have investigated whether some universal structures of maps lift to weighted Hurwitz numbers, such that the topological recursion (it does, for oriented, double, weighted Hurwitz numbers), and the passage from the KP to the BKP hierarchy for some unoriented weighted Hurwitz numbers (like monotone ones). The other generalization of bipartite maps we considered is colored triangulations in dimensions three and higher. They provide a nice meeting ground for topology and combinatorics, where universality classes above dimension 2 can be investigated. In particular, we found that the gluings of 3-balls which maximize the number of edges at fixed number of tetrahedra are in bijection with trees. In even dimensions however, we found that more universality classes can be reached depending on the choice of building blocks. We also wondered whether some of the universal structures featured by maps lift to higher dimensions. In particular, we proved instances of the scheme decomposition in three-dimensional models and maps decorated with crossing loops, and we proved the blobbed topological recursion with only mild assumptions.