The geometry of geometries: matroid theory, old and new

  • Federico Ardila-Mantilla

    San Francisco State University, San Francisco, CA, USA, and Universidad de Los Andes, Bogotá, Colombia
The geometry of geometries: matroid theory, old and new cover
Download Chapter PDF

This book chapter is published open access.


The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total positivity. Matroids capture the combinatorial essence that these different settings share.

In recent years, the (classical, polyhedral, algebraic, and tropical) geometric roots of the field have grown much deeper, bearing new fruits. We survey some recent successes, stemming from three geometric models of a matroid: the matroid polytope, the Bergman fan, and the conormal fan.