Lecture 1: Matroid theory and Tutte's polynomial
A matroid is a combinatorial structure that can be defined by keeping the main
'set properties' of linear dependency in vector spaces. A matroid is naturally associated with a finite set of points (or hyperplanes), and captures the incidence relation (alignment, coplanarity) among these points.
In this lecture, we will give an introduction to the theory of matroids. We will present classical notions and constructions, and give examples and applications. We will then discuss properties of Tutte's polynomial (a polynomial naturally associated with a matroid) and its applications to different topics: Ehrhart's polynomial, knot theory, graphs, etc.
Lecture 2: Ideals and simplicial complexes of matroids
In this lecture we will explain how an ideal IM can be naturally associated with a matroid M. We will discuss an old conjecture due to N. White stating that IM is generated by quadratics corresponding to symmetric base exchanges. We will also study the property of complete intersection of IM and its relation with minors of M. Finally, we will discuss a long-staying conjecture due to Stanley stating that the h-vectors of simplicial complexes of matroids are O-sequence pure.
Lecture 3: Theory of oriented matroids and convexity
An oriented matroid is a combinatorial structure, close to matroids, that captures the convexity relation
of a set of points by taking into account, this time, the signs of the linear dependency relations.
In this lecture, we will give an introduction to the theory of oriented matroids. We will present classical notions and properties, in particular, we study their topological representations. We will also discuss applications to several convexity problems: polytope projective transformation and Gale diagrams, Radon partitions, Hadwiger- and Helly-type problems.
Bibliography
N. White (ed.), Theory of matroids, Cambridge University Press, (1986).
J. G. Oxley, Matroid theory, Oxford Science Publications, (1992).
N. White (ed.), Matroid applications, Cambridge University Press, (1992).
Lecture 4: Semigroups, Frobenius' number and Moebius function
After introducing the semigroups and the diophantine Frobenius problem, we will discuss in this lecture
the Moebius function of a poset naturally associated wth semigroups. We shall explain how the Hilbert series of a semigroup allows to calculate the Moebius function. The latter yields a method to compute, among others, the classical (arithmetic) Moebius function.
Bibliography
J. Ramírez Alfonsín, The diophantine Frobenius problem, Oxford University Press (2005).