Blow-up via parabolic gluing

We will present some recent results on the construction of blow-up solutions for critical parabolic problems of geometric flavor. Initiated in the recent years, the inner/outer parabolic gluing is a very versatile parabolic version of the well-known Lyapunov-Schmidt reduction in elliptic PDE theory. The method allows to prove rigorously some formal matching asymptotics (if any available) for several PDEs arising in porous media, geometric flows, etc….I will give an overview of the strategy and will present several applications to (variations of) the harmonic map flow, Yamabe flow and Yang-Mills flow. I will also present some open questions.