A famous conjecture by Michel stated that all simple Riemannian manifolds are boundary rigid. In this talk, we will first introduce Gromov's filling minimality and its relation to the boundary rigidity problem. Then we will introduce Burago-Ivanov's approach to prove both filling minimality and boundary rigidity for almost Euclidean and almost hyperbolic metrics. If time permits, I will briefly explain how to generalize their argument to almost rank-1 metrics of non-compact type. Speaker(s): Yuping Ruan (U Michigan)