Matroids

Jeffrey Giansiracusa - email: jeffrey.giansiracusa@durham.ac.uk

Description

A matroid is an object that lives in the world of combinatorics and discrete mathematics.  Matroids have many facets.  From one perspective, they look like abstractions of matrices, but from a different angle they look like abstractions of graphs.  And there are many other perspectives.  Matroids are related to certain types of optimisation problems and they have a way of popping up in all sorts of unexpected places.  

There are many different but equivalent ways to define what a matroid is, and getting started in the subject requires relatively little mathematical background.  In this project we will start by exploring the various equivalent axiom systems for matroids and then proceed to topics such as matroid duality and connections with classical theorems of Euclidean and projective geometry.  Depending on interest, the project could then evolve in one of several directions, including connections with tropical geometry, algebra, and rigidity of structures (to name just a few possibilities).

Prerequisites and corequisites

This project has no strict prerequisites, but Algebra II (MATH2581) would be beneficial.

Resources

James Oxley, What is a matroid?, http://www.math.lsu.edu/~oxley/survey4.pdf

Neil White, Matroid applications, Cambridge Univesity Press, 1992.  (Available online through Bill Bryson)

Dominic Welsh, Matroid theory, London Mathematical Society monographs, vol 8, Academic Press, London (1976)