(1) Some results on the point-wise gradient estimate for all positive solutions of the conjugate heat equation. This contrasts to G. Perelmanâ€™s point-wise gradient estimate which works mainly for the fundamental solution rather than all solutions; and some applications to this gradient estimate (Harnack inequality etc.) (some technique may involve the general maximum principle, etc.)

(2) Some result for Sobolev embedding on the complete non-compact manifold under Ricci flow. It would involve some technique like the classic heat kernel estimates, log-Sobolev inequality, Marcinkiewicz interpolation Theorem etc.