Spin glasses are the most basic examples of disordered systems of statistical mechanics with mean-field interactions. The infinite-volume limit of their free energies are described by the celebrated Parisi formula. I will describe a connection between this result and certain Hamilton-Jacobi equations. The talk will then mostly focus on a simpler setting arising from the problem of infering a large rank-one matrix, in which the corresponding Hamilton-Jacobi equation is posed in a finite-dimensional space.