We review recent results of emergent geometries in the BMN matrix model, a one-dimensional gauge theory considered as a non-perturbative formulation of M-theory on the plane-wave geometry. A key to understand the emergent geometries is the eigenvalue distribution of a BPS operator. Gauge-theory calculation shows that the BPS operator reproduces the corresponding supergravity solutions in the gauge/gravity duality and also brane geometries in the M-brane picture. At finite temperatures, these geometries should be realised in a non-trivial way. Monte Carlo simulations of this gauge theory revealed two types of phase transitions: the confinement/deconfinement transition and the Myers transition, which provide insights into the emergence of the geometries. Especially, the numerical results qualitatively agree with the critical temperature of the confinement/deconfinement transition predicted on the gravity side.