Core-collapse supernovae are a candidate source of gravitational waves. They present an opportunity to learn about the process of formation of neutron stars, the properties of matter at high densities and the explosion mechanism itself. This information could be encoded in the frequencies of the oscillations of the newly formed proto-neutron star and its convective motions. This talk will focus on the nature of this oscillations and the mathematical challenges associated to the problem of classifying oscillation modes, which is an important piece to develop functional asteroseismology of proto-neutron stars.