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Pure Yang-Mills amplitudes with all external gluons carrying positivehelicity, known as all-plus amplitudes, have an especially simple structure.The tree amplitudes vanish and, up to at least two loops, the loop-levelamplitudes are related to those of $\mathcal{N}=4$ super-Yang-Mills (SYM)theory. This makes all-plus amplitudes a useful testing ground for new methodsof simplifying more general classes of amplitudes. In this thesis we considerthree new approaches, focusing on the structure before integration.

A Tale of Two Loops: Simplifying All-Plus Yang-Mills Amplitudes