Glass and jamming transitions: From exact results to finite-dimensional descriptions

5 figures, 26 pagesInternational audienceDespite decades of work, gaining a first-principle understanding of amorphous materials remains an extremely challenging problem. However, recent theoretical breakthroughs have led to the formulation of an exact solution in the mean-field limit of infinite spatial dimension, and numerical simulations have remarkably confirmed the dimensional robustness of some of the predictions. This review describes these latest advances. More specifically, we consider the dynamical and thermodynamic descriptions of hard spheres around the dynamical, Gardner and jamming transitions. Comparing mean-field predictions with the finite-dimensional simulations, we identify robust aspects of the description and uncover its more sensitive features. We conclude with a brief overview of ongoing research