Randomized linear algebra has shown that many matrix approximation tasks can be solved efficiently by sampling and sketching, yet deterministic rank revealing methods remain a central step. In this talk, we examine recent theoretical results that clarify what greedy pivoting can and cannot guarantee in rank-revealing factorizations. We first describe a local maximum-volume perspective on pivoted Gaussian elimination and QR factorizations, which yields sharp conditions for when deterministic pivoting reliably reveals numerical rank. We then contrast this with pivoted Cholesky factorizations applied to matrices arising from smooth kernels, showing that greedy pivoting cannot exhibit Kahan-like behavior.