  The four dimensional $\mathcal{N}=4$ super-Yang-Mills (SYM) theory exhibits
rich dynamics in the presence of codimension-one conformal defects. The new
structure constants of the extended operator algebra consist of one-point
functions of local operators which are nonvanishing due to the defect insertion
and carry nontrivial coupling dependence. We study an important class of
half-BPS superconformal defects engineered by D5 branes that share three common
directions with the D3 branes and involve Nahm pole configurations for the SYM
fields on the D3 brane worldvolume. In the planar large $N$ limit, we obtain
non-perturbative results in the 't Hooft coupling $\lambda$ for the defect
one-point functions of both BPS and non-BPS operators, building upon recent
progress in localization and integrability methods. For BPS operator insertions
in the SYM with D5-brane type boundary or interface, we derive an effective two
dimensional defect-Yang-Mills (dYM) theory from supersymmetric localization,
which gives an efficient way to extract defect observables and generates a
novel matrix model for the defect one-point function. By solving the matrix
model in the large $N$ limit, we obtain exact results in $\lambda$ which
interpolate between perturbative Feynman diagram contributions in the weak
coupling limit and IIB string theory predictions on $AdS_5\times S^5$ in the
strong coupling regime, providing a precision test of AdS/CFT with interface
defects. For general non-BPS operators, we develop a non-perturbative
bootstrap-type program for integrable boundary states on the worldsheet of the
IIB string theory, corresponding to the interface defects in the planar SYM.
Such integrable boundary states are constrained by a set of general consistency
conditions for which we present explicit solutions that reproduce and extend
the known results at weak coupling from integrable spin-chain methods.
