  These are lecture notes intended to supplement my second lecture at the
Current Developments in Mathematics conference in 2014. In the first half of
article, we give an introduction to the totally nonnegative Grassmannian
together with a survey of some more recent work. In the second half of the
article, we give a definition of a Grassmann polytope motivated by work of
physicists on the amplituhedron. We propose to use Schubert calculus and
canonical bases to replace linear algebra and convexity in the theory of
polytopes.
