Mean field type equations and some related problems

Abstract:

In this talk, we will first review the resolution of the celebrated Chang-Yang’s conjecture on a sharp Moser-Trudinger type inequality on the two-dimensional sphere. Then we recall the complete theory in dimension one and emphasize the key role played by mean field type equations. Moreover, higher dimensional analogues will be presented and some very recent progress will be introduced. We will also mention some related open problems that the audience might be interested in.