Matroids axiomatise the notion of linear dependence for a list of vectors. While the applications to computer science and optimization are known since long time, surprising interactions of matroid theory with algebra and algebraic geometry were recently discovered, leading to the proof of important combinatorial conjectures and to the introduction of new matroids invariants. In the first part of the talk I will give an elementary introduction to the topic, focusing on examples arising from graphs and from families of hyperplanes in a vector space.
In the second part of the talk I will show that the set of (isomorphism classes of) matroids has a natural structure of Hopf algebra. Then I will introduce a class of matroid-like objects, called minor systems, and describe the related bialgebras. This machinery allows to give rise to a wide number of invariants, old and new.
(Partially based on joint work with Alex Fink and Clement Dupont)