The Bergman complex of a matroid and phylogenetic trees

TL;DR

This paper explores the combinatorial structure of the Bergman complex of a matroid, revealing its relation to the lattice of flats and establishing a homeomorphism with the space of phylogenetic trees.

Contribution

It provides a purely combinatorial description of the Bergman complex and links it to phylogenetic tree space, enhancing understanding in algebraic geometry and combinatorics.

Findings

A subdivision of the Bergman complex corresponds to the order complex of the lattice of flats.

The Bergman fan of the graphical matroid of K_n is homeomorphic to the space of phylogenetic trees T_n.

The work bridges combinatorial geometry and phylogenetics through matroid theory.

Abstract

We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. In addition, we show that the Bergman fan B'(K_n) of the graphical matroid of the complete graph K_n is homeomorphic to the space of phylogenetic trees T_n.