Bergman Fan Research Articles

The Bergman complex of a matroid and phylogenetic trees

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 the proper part 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 × R . This leads to a proof that the link of the origin in T n is homeomorphic to the order complex of the proper part of the partition lattice Π n .

Subdominant Matroid Ultrametrics

Given a matroid M on the ground set E, the Bergman fan e B(M), or space of M- ultrametrics, is a polyhedral complex in R E which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situ- ation, we study the following problem: Given a point ω in R E , we wish to nd an M-ultrametric which is closest to it in the '¥-metric. The solution to this problem follows easily from the existence of the subdominant M-ultramet- ric: a componentwise maximum M-ultrametric which is componentwise smaller than ω. A pro- cedure for computing it is given, which brings together the points of view of matroid theory and tropical geometry. When the matroid in question is the graphical matroid of the complete graph Kn, the Bergman fan e B(Kn) parameterizes the equidistant phylogenetic trees with n leaves. In this case, our results provide a conceptual explanation for Chepoi and Fichet's method for computing the tree that most closely matches measured data.