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A self-consistent exposition of the theory of tree-level superamplitudes ofthe 4d N=4 and 6d N=(1,1) maximally supersymmetric Yang-Mills theories isprovided. In 4d we work in non-chiral superspace and construct thesuperconformal and dual superconformal symmetry generators of the N=4 SYMtheory using the non-chiral BCFW recursion to prove the latter. In 6d weprovide a complete derivation of the standard and hidden symmetries of thetree-level superamplitudes of N=(1,1) SYM theory, again using the BCFWrecursion to prove the dual conformal symmetry. Furthermore, we demonstratethat compact analytical formulae for tree-superamplitudes in N=(1,1) SYM can beobtained from a numerical implementation of the supersymmetric BCFW recursionrelation. We derive compact manifestly dual conformal representations of thefive- and six-point superamplitudes as well as arbitrary multiplicity formulaevalid for certain classes of superamplitudes related toultra-helicity-violating massive amplitudes in 4d. We study massive treesuperamplitudes on the Coulomb branch of the N=4 SYM theory from dimensionalreduction of the massless superamplitudes of the six-dimensional N=(1,1) SYMtheory. We exploit this correspondence to construct the super-Poincare andenhanced dual conformal symmetries of massive tree superamplitudes in N=4 SYMtheory which are shown to close into a finite dimensional algebra of Yangiantype. Finally, we address the fascinating possibility of uplifting massless 4dsuperamplitudes to 6d massless superamplitudes proposed by Huang. We confirmthe uplift for multiplicities up to eight but show that finding the uplift ishighly non-trivial and in fact not of a practical use for multiplicities largerthan five.

From Six to Four and More: Massless and Massive Maximal Super Yang-Mills Amplitudes in 6d and 4d and their Hidden Symmetries