Scattering Amplitudes, Unitarity and the Positve Grassmannian

In recent years, a complete reformulation of perturbative quantum field theory has been found for certain theories. Although a similar reformulation is likely to exist for any quantum field theory, it is exceptionally simple in the case of planar, maximally supersymmetric Yang-Mills. I will (briefly) review this new formulation, describing the connections between certain field-theoretic quantities called `on shell diagrams' and a space known as the `positive Grassmannian', how both can be characterized combinatorially, and how the entire S-matrix can be defined in terms of these combinatorial objects. I will then describe how these ideas can be applied to teach us about multi-loop amplitudes, and some surprising new features that appear at 2-loops and beyond, even for planar N=4 super Yang-Mills.