Parole chiave

• finite metric space
• fundamental polytopes
• hyperplane arrangement
• matroid
• weighted graph

Abstract Motivated by questions from computational biology, we tackle the problem of a combinatorial classification of finite metric spaces by means of a new polyhedral invariant introduced by Vershik in 2010: the metric space’s “fundamental polytopes”. These originate from the theory of optimal transport (where they are often named after Wasserstein or Kantorovich-Rubinstein) and have recently found applications in a host of different contexts, from algebraic statistics to tropical geometry to the theory of reaction networks. Nevertheless, the most basic questions on their structure remain to date unanswered.

In this talk I will begin by defining the fundamental polytopes of finite metric spaces and sketching the motivation for our work. I will then show how matroid theory allows to describe the combinatorial structure of the fundamental polytopes associated to tree-like metric spaces. I will also discuss some partial results for the case of a special type of phylogenetic networks and, time permitting, I will also present some lines of current research.