Phased matroids are combinatorial objects that play the same role for the complex numbers that oriented matroids play for the real numbers. They were introduced in 2012 as a potential tool for various problems in geometry and topology. I'll discuss the state of the field, focusing on progress toward a combinatorial model for the Grassmannian G(r,Cⁿ).

(I talked about the same material last spring, but the ideas have been clarified since then.)