Scientific Research and Writing: Representation Theory of Matroids

In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. After an introduction covering different equivalent definitions of matroids, we will focus on the representation theory of matroids: given a matroid M and a field k one asks whether there exists a tuple of elements in a k-vector space obeying exactly the linear dependencies prescribed by M. Matroid representations are analogous to group representations; both types of representation provide abstract algebraic structures (matroids and groups respectively) with concrete descriptions in terms of linear algebra. It turns out that matroid representations are best understood by considering representations over so-called tracts which are algebraic structures generalizing fields. This is, for example, due to the fact that, for every matroid M, the functor from the category of tracts to sets that takes a tract T to the set of representations of M over T is always representable.

Knowledge in mathematics at the Bachelor's level is required. Basic knowledge in category theory and algebraic geometry is helpful.