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 inferring a large rank-one matrix, in
which the corresponding Hamilton-Jacobi equation is posed in a
finite-dimensional space.