  We use numerical bootstrap techniques to study correlation functions of
scalars transforming in the adjoint representation of $SU(N)$ in three
dimensions. We obtain upper bounds on operator dimensions for various
representations and study their dependence on $N$. We discover new families of
kinks, one of which could be related to bosonic QED${}_3$. We then specialize
to the cases $N=3,4$, which have been conjectured to describe a phase
transition respectively in the ferromagnetic complex projective model $CP^2$
and the antiferromagnetic complex projective model $ACP^{3}$. Lattice
simulations provide strong evidence for the existence of a second order phase
transition, while an effective field theory approach does not predict any fixed
point. We identify a set of assumptions that constrain operator dimensions to
small regions overlapping with the lattice predictions.
