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In this talk,we prove Li-Yau-Hamilton Harnack inequalities and derive the Perelman W-entropy formula for the heat equation of the Witten Laplacian on 29 Riemannian manifolds.We also prove a rigidity theorem for the W-entropy and a splitting theorem for the Kaimanovich entropy associated with the Witten Laplacian.Finally,if the time is enough,we will use the gradient flow of the Voiculescu entropy and the optimal transportation theory to derive the uniqueness of the McKean-Vlasov equation,and to prove the Law of Large Numbers for the Generalized Dyson Brownian motion in random matrices theory.