Algebra and geometry of the scattering equations

Four years ago, Cachazo, He and Yuan found a system of algebraic equations, now named the "scattering equations", that effectively encoded the kinematics of massless particles in such a way that the scattering amplitudes, the quantities of physical interest, in gauge theories and in gravity could be written as sums of rational functions over their solutions. Without assuming any background in scattering amplitudes or quantum field theory, I shall aim to discuss the solution and properties of this rather remarkable system, reformulated as complex polynomial equations, and, in particular, how the quantities of interest are determined by the zero-dimensional variety characterized by the scattering equations.