# Special Year Seminar II

## Foundations of Matroids

The second lecture features the nuts and bolts of the invariants from first lecture, which we call foundations. We explain the structure theorem for foundations of ternary matroids, which is rooted in Tutte's homotopy theorem. We show how this theorem implies a series of results about the representation theory of ternary matroids.

As another application of foundations, we demonstrate a short proof of a folklore theorem (attributed to Laurent Lafforgue) about the realization space of a rigid matroid. Yet another application passes through the relation between Lorentzian polynomials with matroids over triangular hyperfields, which lets us answer (negatively) conjectures/questions of Petter Branden about the topological type of the spaces of Lorentzian and stable polynomials.