Motivations for and applications of Matroid Theory?

Question

I have taken an interest in this topic recently. If one is unfamiliar with matroids, I will give the definition here.

Let $M=(E,\mathcal I)$ where $E$ is a finite set called the ground set and $\mathcal I$ is a collection of subsets of $E$ called the independent sets satisfying three properties:

1. The empty set is independent, i.e. $\varnothing \in \mathcal I$.
2. Every subset of an independent set is independent, i.e. if $I\in\mathcal I$ and $J\subset I$, then $J\in\mathcal I$.
3. If $I,J\in\mathcal I$ such that $|I|>|J|$, there exists an element $x\in I$ such that $J\cup\{x\}\in\mathcal I$.

The third property is known as the augmentation or exchange property and of course what makes this definition interesting. Matroids are a structure that abstracts and generalizes the notion of linear independence in vector spaces.

Now, I am interested in the general theory of matroids, and am using these video lectures as well as Oxley's Matroid Theory to study the topic. However, I am having some difficulty finding motivation as to why these objects are interesting and how they can be applied to other fields of mathematics. My understanding is that this is largely a combinatorial topic, which is one of my weaker areas in mathematics (I am strongest in analysis and topology).

I would appreciate it if someone could provide motivation as to why matroids are an interesting topic to study, as well as ways they can be linked and applied to other fields of mathematics, especially topology. I would also appreciate other references, as it seems this topic is not as widely studied as, say, graph theory or enumerative combinatorics.

Answer 1

A motivation of Whitney's for introducing matroids was to clarify what duality would mean when a graph is not planar. Independently (pun intended?) of Whitney, Takasawa's motivation was to study independence of axioms in an axiom system, with projective geometry in mind.

Kung's A Source Book in Matroid Theory has a chapter dedicated to the origins of matroids.

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Applications (not even close to being an exhaustive list)

Algebraic and computational geometry. Let $R=k[x_1,\dots,x_n]$ be a polynomial ring over a field of characteristic zero. Let $P$ be a prime ideal of $R$. The algebraic matroid $\mathcal A(P)$ of $P$ is the collection of subsets $S$ of $\mathcal S=\{x_1,\dots,x_n\}$ such that $$P\cap k[S]=\{0\},$$ where $k[S]$ is the subring of $R$ on the indeterminates in $S$.

In particular, the dimension of $P$ matches the size of the bases of $\mathcal A(P)$. Very interesting are the circuits of $\mathcal A(P)$, i.e. subsets $C\subseteq\mathcal S$ such that $P\cap k[C]\neq\{0\}$ and $P\cap k[C']=\{0\}$ for any $C'\subset C$. The ideal $P\cap k[C]$ (sometimes called an elimination ideal) is generated by a single irreducible polynomial, called a circut polynomial, unique up to multiplication with a non-zero element of $k$.

A suitable setting is that of Elimination theory and Gröbner bases.

See Rosen, Sidman, Theran - Algebraic Matorids in Action, https://arxiv.org/pdf/1809.00865.pdf, for an introduction

Rigidity theory. Consider a point configuration $P$ of $n>3$ points in $\mathbb R^2$. Let $d_{ij}$ denote the square of the distance between from point $i$ to $j$. Let $R=\mathbb C[\{d_{ij}\mid 1\leq i<j\leq n\}]$ and $CM$ the ideal generated by the $5\times 5$ minors of the $(n+1)\times (n+1)$ matrix obtained from the symmetric matrix $D=[d_{ij}]$ with $D_{ii}=0$ by adjoining to the top and to the left a row and a column $(1,1,\dots,1)$ of length $(n+1)$.

The ideal $\text{CM}$ is called the Cayley-Menger ideal. Consider the algebraic matroid $\mathcal A(\text{CM})$ of this ideal. It is isomorphic to the rigidity matroid of the point configuration $P$. Its bases are the so-called Laman graphs, i.e. minimaly rigid subgraphs of the complete graph $K_n$ on $n$ vertices.

Rigidity theory has wide applications in engineering, biology, chemistry, robotics etc. One of the most important open problems in the field is to find a combinatorial characterisation of minimally rigid graphs of point configurations in $\mathbb R^3$. A hope is that there is a matroid whose bases would correspond to the minimally rigid graphs in 3-space.

Whiteley - Some matroids from discrete applied geometry

Graver, Servatius, Servatius - Combinatorial Rigidity.

Topology. Matroids give a stratification of the (n,k)-Grassmannian. However, a much more interesting object to study with topology in mind are oriented matroids. These are matroids enriched with an abstract concept of orientation.

A highlight of oriented matroid theory is the combinatorial formula for the Pontryagin classes of a triangulated manifold, obtained by Gelfand and MacPherson.

Gelfand, MacPherson - Combinatorial formula for the Pontrjagin classes

An idea of MacPherson was to study approximations of differential manifolds via combinatorial differential manifolds for which tangent bundles are replaced with oriented matroid bundles.

MacPherson - Combinatorial Differential Manifolds

Laura Anderson - Topology of Combinatorial Differential Manifolds

A recent major advancement was the introduction of Hodge theory for matroids by Adiprasito, Huh and Katz. For a short overview, see

Adiprasito, Huh, Katz - Hodge Theory for Matroids.

Semi-algebraic geometry. A fundamental result was provided by Mnëv in his Universality Theorem stating that semi-algebraic subsets in $\mathbb R^n$ over the integers are stably equivalent to realisation spaces of oriented matroids.

Mnëv - The universality theorems on the classification problem of configuration varieties and convex polytopes varieties

Vakil - Murphy's Law in algebraic geometry: Badly-behaved deformation spaces

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There are more applicatons in e.g. tropical geometry, Coxeter groups and combinatorial optimization.

Some general references:

Björner, Las Vergnas, Sturmfels, White, Ziegler - Oriented Matroids

Borovik, Gelfand, White - Coxeter matroids

E Katz - Matroids for algebraic geometers