What is a matroid? Theory and Applications, from the ground up

Gian-Carlo Rota said that “Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day.” [20] Hassler Whitney introduced the theory of matroids in 1935 and developed a striking number of their basic properties as well as different ways to formulate the notion of a matroid. As more and more connections between matroid theory and other fields have been discovered in the ensuing decades, it has been realized that the concept of a matroid is one of the most fundamental and powerful in mathematics. Examples of matroids arise from networks, matrices, configurations of points, arrangements of hyperplanes, and geometric lattices; matroids play an essential role in combinatorial optimization. We all know some matroids, but not always by name. In mathematics, notions of independence akin to linear independence arise in various contexts; matroids surface naturally in these situations. We provide a brief, accessible introduction so that those interested in matroids have a place to start. We look at connections between seemingly unrelated mathematical objects, and show how matroids have unified and simplified diverse areas. An introduction to matroids can be found in An Introduction to Matroid Theory [28], in the AMS Feature Column, Matroids: The Value of Abstraction [17], and in Matroids You Have Known [18]. The two books entitled Matroid Theory [19] and [22] provide a strong foundation, as does the series Theory of Matroids [26], Matroid Applications [25], and Combinatorial Geometries [24]. Many of the key early papers are reprinted in A source book in matroid theory [14] with illuminating commentaries. Early work on matroids [8] can be found by H. Whitney [27], G. Birkhoff [2], S. Maclane [16], and B.L. van der Waerden [21]. For background in graph theory, see the graph theory books of West [23], Diestel [9], Wilson [29], or Harary [12]. For background in combinatorics, see Introductory Combinatorics [4] or [7]. For applications of matroids, see Combinatorial Optimization: Networks and Matroids [15], the three chapters on matroid theory in Handbook of Combinatorics [10], and Matroid Applications [25]. For some connections of matroids to other areas of discrete mathematics, see Discrete and Combinatorial Mathematics: An Applied Introduction [11] and The Many Names of (7,3,1) [5].