Tale of two loops: simplifying all-plus Yang-Mills amplitudes

Pure Yang-Mills amplitudes with all external gluons carrying positive helicity, known as all-plus amplitudes, have an especially simple structure. The tree amplitudes vanish and, up to at least two loops, the loop-level amplitudes are related to those of N = 4 super-Yang-Mills (SYM) theory. This makes all-plus amplitudes a useful testing ground for new methods of simplifing more general classes of amplitudes. In this thesis we consider three new approaches, focusing on the structure before integration. We begin with the planar (leading-colour) sector. A D-dimensional local-integrand presentation, based on four-dimensional local integrands developed for N = 4 SYM, is developed. This allows us to compute the planar six-gluon, two-loop all-plus amplitude. Its soft structure is understood before integration, and we also perform checks on collinear limits. We then proceed to consider subleading-colour structures. A multi-peripheral colour decomposition is used to find colour factors based on underlying tree-level amplitudes via generalised unitarity cuts. This allows us to find the integrand of the full-colour, two-loop, five-gluon all-plus amplitude. Tree-level BCJ relations, satisfied by amplitudes appearing in the cuts, allow us to deduce all the necessary non-planar information for the full-colour amplitude from known planar data. Finally, we consider representations satisfying colour-kinematics duality. We discuss obstacles to finding such numerators in the context of the same five-gluon amplitude at two loops. The obstacles are overcome by adding loop momentum to our numerators to accommodate tension between the values of certain cuts and the symmetries of certain diagrams. Control over the size of our ansatz is maintained by identifying a highly constraining, but desirable, symmetry property of our master numerator