Morse theory provides a framework for studying the infinite-dimensional space of embedded hypersurfaces in a fixed ambient manifold through the analysis of the area functional. A special role in this picture is played by minimal hypersurfaces, which are precisely the critical points of this functional. By analogy with classical Morse theory, one expects that a minimal hypersurface of Morse index I should admit an I-dimensional unstable manifold emanating from it. In this talk, I will present recent results that make this heuristic precise by constructing and characterizing ancient solutions to mean curvature flow that emanate from minimal hypersurfaces. Particular focus will be given to the case of hypersurfaces with boundary.
