  In this thesis, we study the integrands of a special four-point correlation
function formed of protected half-BPS operators and scattering amplitudes in
planar supersymmetric $\mathcal{N}=4$ Yang-Mills. We use the `soft-collinear
bootstrap' method to construct integrands of the aforementioned correlator and
four-point scattering amplitudes to eight loops. The result is then extended to
ten loops, by introducing two graphical relations, called the `triangle' and
`pentagon' rules. These relations provide consistency conditions on the
coefficients, and when combined with the `square' rule, prove sufficient to fix
the answer to ten loops. We then proceed to study the correlator/amplitude
duality by taking six and seven adjacent points of the four-point correlator to
be light-like separated. A conformal basis (with rational coefficients) is used
to extract amplitude integrands for both six and seven particles up to two
loops - more precisely, the complete one-loop amplitude and parity-even
two-loop amplitude (at two loops, we use a refined prescriptive basis). We also
construct an alternative six-point one-loop basis involving integrands with
conformal cross-ratio coefficients, and reverse the duality to algebraically
extract integrands from an ansatz, by introducing the Gram determinant.
