Optimal stability for Hardy-Littlewood-Sobolev and fractional Sobolev inequalities

In this talk, we will discuss some recent works on sharp stability for some important geometric and functional inequalities. These include asymptotically sharp lower bounds for the stability for the Hardy-Littlewood-Sobolev and fractional Sobolev inequalities in Euclidean spaces, the stability for the Sobolev inequality on the Heisenberg group where the rearrangement inequality is absent. We recently develop a rearrangement-free argument to establish the stability for such Sobolev inequality.