Transitive factorizations of permutations and geometry

TL;DR

This paper explores transitive factorizations of permutations and their connections to various mathematical fields, revealing a unifying algebraic combinatorics perspective that links graph theory, random matrices, and moduli spaces.

Contribution

It provides a comprehensive account of transitive permutation factorizations and demonstrates their impact across multiple areas of mathematics, highlighting new unifying insights.

Findings

Connections between permutation factorizations and graph embeddings

Applications to random matrices and branched covers

Unifying algebraic combinatorics framework

Abstract

We give an account of our work on transitive factorizations of permutations. The work has had impact upon other areas of mathematics such as the enumeration of graph embeddings, random matrices, branched covers, and the moduli spaces of curves. Aspects of these seemingly unrelated areas are seen to be related in a unifying view from the perspective of algebraic combinatorics. At several points this work has intertwined with Richard Stanley's in significant ways.