  We investigate the singularity structure of the $(-1)^F$ graded partition
function in QCD with $n_f \geq 1$ massive adjoint fermions in the large-$N$
limit. Here, $F$ is fermion number and $N$ is the number of colors. The large
$N$ partition function is made reliably calculable by taking space to be a
small three-sphere $S^3$. Singularites in the graded partition function are
related to phase transitions and to Hagedorn behavior in the $(-1)^F$-graded
density of states. We study the flow of the singularities in the complex
"inverse temperature" $\beta$ plane as a function of the quark mass. This
analysis is a generalization of the Lee-Yang-Fisher-type analysis for a theory
which is always in the thermodynamic limit thanks to the large $N$ limit. We
identify two distinct mechanisms for the appearance of physical Hagedorn
singularities and center-symmetry changing phase transitions at real positive
$\beta$, inflow of singularities from the $\beta=0$ point, and collisions of
complex conjugate pairs of singularities.
