The totally positive (and totally non negative) k,n-Grassmannian is the space of all k dimensional
linear subspaces of the n-dimensional Euclidean space whose Plücker coordinates are all positive
(non-negative respectively). It was studied by Postnikov, who gave a cell decomposition of the non
negative Grassmannian whose cells are in bijection with a plethora of combinatorial objects, most
notably positroids. The totally positive Grassmannian gained much attention after several connections
to physics were established, including solutions to the KP equation for shallow water waves and
scattering amplitudes in quantum field theory. In the latter, Feynman diagrams in the supersymetric
N=4 Yang Mills theory are modeled by plabic graphs.
In this talk we will define plabic graphs and show how they can be used to study and answer several
questions about the totally positive Grassmannian. Among such questions are: which sets of Plücker
coordinates can vanish? How many Plücker coordinates have to be tested to decide total positivity?
How to can we parametrize cells of the totally non-negative Grassmannian? What is its topology?