Georgia Institute of Technology College of Sciences

Matroids are widely used objects in combinatorics; they arise naturally in many situations featuring vector configurations over a field. But in some contexts the natural data are elements in a module over some other ring, and there is more than simply a matroid to be extracted. In joint work with Luca Moci, we have defined the notion of matroid over a ring to fill this niche. I will discuss two examples of situations producing these enriched objects, one relating to subtorus arrangements producing matroids over the integers, and one related to tropical geometry producing matroids over a valuation ring. Time permitting, I'll also discuss the analogue of the Tutte invariant.