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.
