Undergrad Mini Course 2: An introduction to matroid theory
Imagine you are visiting Honolulu and have a packed schedule of activities. Your map shows a dot for every site to visit, the best poke spot, and a secret bay to snorkel with sea turtles. Using the roads connecting these dots as edges of a graph, you wish to find all the minimal routes connecting these destinations that avoid forming a cycle. A matroid holds the key! In this mini-course we will get our hands dirty defining, computing, and exploring various perspectives of matroids. We will narrow in on realizable matroids, those arising from linear systems, and in particular, a well-behaved family of realizable matroids called Positroids. By exploring the many combinatorial objects associated with Positroids, we will touch briefly on their far-reaching implications in other areas of mathematics and science. This is geared towards undergraduate students and will assume some familiarity with linear algebra.
Link to presentation slides: https://www.academia.edu/40810771/Modernmath2019_minicourse