The positive Dressian and the amplituhedron

The Dressian and the tropical Grassmannian parameterize abstract and realizable tropical linear spaces; but in general, by work of Herrmann-Jensen-Joswig-Sturmfels, the Dressian is much larger than the tropical Grassmannian. There are natural positive notions of both of these spaces -- the positive Dressian, and the positive tropical Grassmannian -- so it is natural to ask how these two positive spaces compare. I'll discuss joint work with David Speyer in which we showed that the positive Dressian equals the positive tropical Grassmannian. This result has several applications, for example a new "tropical" proof of da Silva's 1987 conjecture that all positively oriented matroids are realizable. In joint work with Lukowski and Parisi, we showed that the positive Dressian controls positroidal subdivisions of the hypersimplex, and furthermore showed that there is a striking parallel between positroidal subdivisions of the hypersimplex, and positroidal subdivisions of the m=2 amplituhedron.