This lecture will discuss some fundamental enumerative invariants of matroids, such as the characteristic and the Tutte polynomial and (at least partly) the face enumerators of some important simplicial complexes which can be associated to them, such as the independence complex, the broken circuit complex and the order complex of the lattice of flats. Applications to the face enumeration of real hyperplane arrangements will be given and some familiar objects of study in graph theory, such as the chromatic polynomial of a graph, will come up as special cases from graphical matroids. The lecture will be accessible to students familiar with only basic notions and constructions of matroid theory.

We will introduce the notion of shellability of a simplicial complex and use it to study some simplicial complexes associated to a matroid (the independence complex, Broken Circuit complex, the No Broken Circuit complex, the order complex of the lattice of flats). This will link the homotopy type of those complexes with known enumerative invariants of matroids. We will close with a recent result characterizing matroids ("cryptomorphically") in terms of a special shellability property.

In this lecture I will try to offer several perspectives on transversal matroids. The starting point will be the the fact, proved by Edmonds and Fulkerson in 1965, that the set of partial transversals of a collection of sets are the independent sets of a matroid. From there we will look at the basic properties of such transversal matroids and we will introduce other useful, equivalent ways of defining and representing them. Our running examples will be two particular classes of transversal matroids, lattice path matroids and bicircular matroids, defined from paths in the plane and from graphs, respectively. In the second part we will go into more specific results related to the questions "How can we tell if a given matroid is transversal?" and "How could we generate all transversal matroids?". We will also survey some research questions that are open for general matroids but solved for transversal matroids or particular subclasses.

In this introductory lecture the basic concepts of matroid theory will be treated. We start from the definition of a matroid and will see several examples, showing the relation with other branches of mathematics like linear algebra, graph theory and combinatorics. Then we have a look at important concepts related to matroids: independent sets, bases, circuits, rank function, closure, flats, and hyperplanes. We will see that each of them can be used to define a matroid, leading to so called "cryptomorphisms". Finally, several ways of making new matroids from old are discussed, such as duality, direct sum, deletion and contraction. All the theory will be accompanied by lots of examples.

TBA

I will consider a basic question in matroid theory: how many matroids are there on a fixed ground set of n elements? And how about asymptotics: which fraction of the matroids on n elements are connected, have a fixed minor, etc. Such questions seem inherently more difficult to answer for matroids than for say, graphs, since matroids are defined by a set of axioms and not given constructively.
In the early seventies, Knuth and Piff showed lower and upper bounds on the number of matroids, but their bounds did not quite match. I will review these early results and several conjectures on matroid asymptotics, and then explain the several recently developed techniques for counting matroids. This will involve elementary combinatorics as well as entropy counting and spectral graph theory.