The Next-to-Simplest Quantum Field Theories

The past few years have seen dramatic breakthroughs in our
understanding of scattering amplitudes in gauge theories and
gravity. These developments led to the conjecture that N=4 Super Yang Mills might have the simplest S-matrix of all quantum field theories despite having a very complicated Lagrangian. In this talk, we will review these developments. We will then describe our search for the ``Next-to-Simplest'' Gauge theories. It turns out that demanding the simplicity of gluon scattering amplitudes leads to linear Diophantine equations involving the higher Casimirs of the matter representations in the theory! The solution of these group theoretic constraints leads to some new theories that have a simple S-matrix but have hitherto not
attracted much attention.