  We reveal properties of global modes of linear buoyancy instability in stars,
characterised by the celebrated Schwarzschild criterion, using non-Hermitian
topology. We identify a ring of Exceptional Points of order 4 that originates
from the pseudo-Hermitian and pseudo-chiral symmetries of the system. The ring
results from the merging of a dipole of degeneracy points in the Hermitian
stablystratified counterpart of the problem. Its existence is related to
spherically symmetric unstable modes. We obtain the conditions for which
convection grows over such radial modes. Those are met at early stages of
low-mass stars formation. We finally show that a topological wave is robust to
the presence of convective regions by reporting the presence of a mode
transiting between the wavebands in the non-Hermitian problem, strengthening
their relevance for asteroseismology.
