Prescribing scalar curvature in conformal geometry

We consider classical the question of prescribing the scalar curvature
of a manifold via conformal deformations of the metric, dating back to
works by Kazdan and Warner, and related to the well-known Yamabe problem.
It is challenging as it involves equations of critical type, leading to possible
blow-ups of solutions. The question is mainly understood in low dimensions,
where blow-ups of solutions are proven to be "isolated simple".
We will present some recent progress in arbitrary dimensions and
for the case of manifolds with boundary, where blow-ups of new type occur.