Real phase structures on matroid fans

In this talk, I will define real phase structures on matroid fans and prove that a real phase structure is cryptomorphic to providing an orientation of the underlying matroid. Real phase structures can be extended to general tropical varieties and we can define the real part. In the matroid setting, this yields the topological representation of an oriented matroid in the sense of Folkman and Lawrence. In addition, the real part determines a homology class in a real toric variety and the conditions for real phase structures can be thought of the real analogues of Minkowski weights of fans. This is joint work in progress with Johannes Rau and Arthur Renaudineau. Lastly, 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. Determining whether a matroid is orientable NP-complete, so determining whether or not the class is non-zero can be expected to be very difficult in general.