Soliton Resolution Conjecture for the energy-critical heat equation

The Soliton Resolution Conjecture predicts that finite-energy solutions to nonlinear dispersive PDEs asymptotically decouple into a sum of stationary solutions, called solitons, and free radiation, with an error that goes to zero in the energy norm. In this talk, we discuss the conjecture for the energy-critical nonlinear heat equation in dimension $d\geq 3$ and present its proof in the radial case. If time permits, we will also discuss recent progress in the non-radial setting, with potential applications to other geometric flows such as the Yang-Mills heat flow.