# Math 580A: Topics in Combinatorial Analysis Matroid Theory Fall 2023

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#### Teacher

Thomas Zaslavsky
Office: WH-216
Email: zaslav@math.binghamton.edu

#### Office hours

My regular office hours are M, W, F: 1:00–3:00. Drop in! You do not need an appointment. If I'm not in my office, I am probably nearby and will be back shortly...
Also by appointment at other times. See me or email to make an appointment.

Class meets on M, W, F at 3:30 in WH-309. I also meet each student individually every week.

#### Course goals

This course is an introduction to the basic concepts and constructions of matroid theory and to the chief examples. This is not an introductory graduate course. The absolute minimum requirement is a good understanding of abstraction and a solid modern algebra background (as from a graduate course in that).

The course is based on:

• James Oxley, Matroid Theory, second edition, Oxford University Press, 2011. (Do not use the first edition.)
The book doesn't cover everything; but it is an exceptionally good graduate textbook.
• Lectures and readings about hyperplane arrangements, chromatic (for graphs) and characteristic (for arrangements) polynomials, and signed, gain, and biased graphs.

### Course Work

I expect you, the students, to study the material and to work on as many of the exercises as you can. I'll collect written work every week or two and return it with comments, for your edufication. There will not be any tests! I'll meet separately with each student frequently to discuss your progress and any questions you may have.

### A Brief Description of Matroid Theory

What is a matroid, really? A matroid is not one thing; it is a complex of properties, no one of them most important. Here is a description of a function, which could be said just as well of a matroid.

A function was not, for Riemann, a mere set of points; still less was it any of its pictorial representations as a graph or a table; and still less a collection of expressions involving algebraic formulas. ... What then, was a function? It was an object, from which none of its attributes could properly be detached. Riemann saw a function the way chess grandmasters are said to see a game, all at once, as a unified whole, a Gestalt.
Prime Obsession by John Derbyshire, 2003, p. 129.

Matroid theory originated as an abstract study of the properties of linear dependence of finite sets of vectors in a vector space. The idea was to forget the details of linear dependence relations and only remember which sets of vectors are linearly dependent. There are some nice axioms to describe a property abstracting linear dependence – let's call it ``dependence'' – that allow one to define (purely abstractly) such basic vector-space properties dimension (we call it ``rank''), linearly closed sets, bases, and much more. The main thing they don't allow one to do is to find the vector space, because the axioms admit examples that cannot possibly come from vectors. A set with a list of dependent subsets (or an equivalent, like a rank function or a list of bases) is called a matroid.

Another origin of matroid theory is in graph theory. There is a natural matroid on a graph: the elements of the matroid are the edges of the graph and the dependent subsets are the edge sets that contain a circuit (a.k.a. cycle or closed path). Many interesting and important graph properties are naturally expressed in terms of matroids. For instance, a spanning tree of a graph is the same as a basis of the graphic matroid. The chromatic polynomial of a graph (this counts proper colorings; it appeared in an attempt to prove the Four-Color Theorem) is another thing that is essentially matroidal, although the connection is less simple. Graph connectivity is closely related to matroids, although here there are some complications.

There are many other sources of matroid theory and examples, including hyperplane arrangements (which have their own analogs of chromatic polynomials), transversals, and whatever.

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