Perturbative Correlation Functions and Scattering Amplitudes in Planar $\mathcal{N}=4$ Supersymmetric Yang-Mills

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.