In this talk, we will report some recent progress on sharp stabilities of some geometric and functional inequalities.
These include the stability problem of the log-Sobolev inequality and Moser-Onofri inequality on the sphere, the explicit lower bounds for the stability of Hardy-Littlewood-Sobolev inequality as well as the stability for the higher and fractional order Sobolev inequalities. The latter extends the recent explicit lower bound result for the stability of the first order Sobolev inequality due to Dolbeault, Figalli, Frank, Esteban and Loss.
These are joint works with Lu Chen and Hanli Tang. Moreover, we will also give the sharp stabilities for a large class of Caffarelli-Kohn-Nirenberg (CKN) inequalities (including the Heisenberg uncertainty principle and Hydrogen uncertainty principle) with best constants in the stability inequalities. This part is joint work with A. Do, C. Cazacu, J. Flynn and N. Lam.

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