Scattering Amplitudes and the Positive Grassmannian

The traditional formulation of quantum field theory is built on two pillars - locality and unitarity. The standard apparatus of Lagrangians and path integrals allows us to make these two fundamental principles manifest. This is however associated with the introduction of a large amount of (gauge) redundancy in our description of physics which also might hide some fundamental symmetries. Many of the recent developments have been fueled by an intensive exploration of N=4 SYM in the planar limit as a laboratory for four-dimensional gauge theories. The all-loop integrand for scattering amplitudes in this theory can be determined using recursion relations in a way that is closely tied to remarkable new structures in algebraic geometry, associated with the contour integral over Grassmannian manifolds. This formulation makes all symmetries of the theory manifest -- in addition to the conformal symmetry there is also long-hidden dual conformal symmetry (which is absolutely invisible in the standard description). If history is any guide, formulating physics in a way that makes the symmetries manifest should play a central role in the story. The Grassmannian picture does exactly this and in our talk we will derive the connection between scattering amplitudes and the Grassmannian starting physically from first principles.