Real phase structures on matroid fans
The tropical geometry of matroid fans has been a recent powerful tool for understanding many matroid invariants. In this talk, I will define real phase structures on fans and prove that a real phase structure on a matroid fan is cryptomorphic to providing an orientation of the underlying matroid. Therefore, we can use tropical techniques to study oriented matroids. We can also define the real part of a fan equipped with a real phase structure. In the matroid setting, this yields the topological representation of an oriented matroid in the sense of Folkman and Lawrence, and is related to results of Ardila-Klivans-Williams and Celaya.
This is joint work with Johannes Rau and Arthur Renaudineau.
Then using real structures I will propose a definition of the first Stiefel-Whitney class of a matroid. This is a homological class which is zero if and only if the matroid is orientable. Thus it is the linguistic analog of the first Stiefel-Whitney class of a manifold in the matroid setting. However, determining whether a matroid is orientable is an NP-complete problem, so determining whether or not the class is non-zero can be expected to be very difficult in general!