Marginal stability in finite dimensions

Glassy materials are omnipresent in everyday life from windows to plastics to piles of sand. Yet our understanding of both their (equilibrium) liquid and (out-of-equilibrium) solid phases lags far behind that of crystalline counterparts. Recent advances are rapidly changing the ways in which we understand these common-yet-physically-enigmatic materials. This talk overviews one such advance -- the discovery of the Gardner phase transition from normal to marginally-stable glasses. Our work in particular indicates that such a transition, first found in abstract infinite-dimensional models, can survive down to the three-dimensional world. This transition reinforces the overriding role of rugged free-energy landscapes that dictate physics of glassy systems, with tangible consequences on jamming, yielding, and beyond.