Uniform density in matroids, matrices and graphs

A matroid M is uniformly dense if rank(M|X)/|M|X|≥rank(M)/|M|for all nonempty restrictions M|X. These matroids are extremal for certain connectivity, packing and covering properties and have applications in the design of robust networks. In this talk, I will discuss a new characterization of uniform density derived from the geometry of matroid polytopes and some of its consequences. As a first application, using the inverse moment map we show that uniformly dense real matroids (i.e. real matrices) are parametrized by a subvariety of the Grassmannian. In the case of positroids, this becomes a linear section with the nonnegative Grassmannian. Second, we show that regular uniformly dense graphic matroids have strong connectivity properties and admit a perfect matching. To conclude, I will mention a number of open problems related to uniform density: some polytopes, positroids and a conjecture.

This is joint work with Raffaella Mulas, available on arxiv.org/abs/2306.15267.