  Self induced neutrino flavor conversions in the dense regions of stellar core
collapse are almost exclusively studied in the standard two flavor scenario.
Linear stability analysis has been successfully used to understand these flavor
conversions. This is the first linearized study of $\textit{three flavor}$ fast
instabilities. The `fast' conversions are fascinating distinctions of the dense
neutrino systems. In the fast modes the collective oscillation dynamics are
independent of the neutrino mass, growing at the scale of the large
neutrino-neutrino interaction strength ($10^5$ km$^{-1}$) of the dense core.
This is extremely fast, in comparison to the usual `slow' collective modes
driven by much smaller vacuum oscillation frequencies ($10^0$ km$^{-1}$). The
three flavor analysis shows distinctive characteristics for both the slow and
the fast conversions. The slow oscillation results are in qualitative agreement
with the existing nonlinear three flavor studies. For the fast modes, addition
of the third flavor opens up possibilities of influencing the growth rates of
flavor instabilities when compared to a two flavor scenario.
