Seminars$W$-entropy formula, Perelman's Ricci flow and optimal transport on manifolds with weighted measure
81
reads
In his seminal paper, Perelman introduced the $W$-entropy for the Ricci flow and proved its monotonicity. This plays a crucial role in the proof of the no local collasping theorem and ”removes the major stumbling block in Hamilton’s approach to geometrization”. In this talk, after a brief review of Perelman's $W$-entropy formula for Ricci flow, I will present our results on the $W$-entropy formula for the heat equation of the Witten Laplacian on manifolds with weighted measure. I will also give a probabilistic interpretation of Perelman's mysterious $W$-entropy for Ricci flow. Then I will present some results on the optimal transport problems for the Fokker-Planck diffusions on manifolds equipped with Perelman's Ricci flow, which can be viewed a natural correspondence of some previous results due to Otto, Lott-Villani, von Renesse-Sturem, McCann-Topping and Lott, etc. We point out that there is an interesting similarity between our $W$-entropy formula for the Witten Laplacian and Lott-Villani and Sturm's result on the monotonicity of the Boltzmann entropy along geodesic on the Wasserstein space over compact Riemannian manifolds. Finally we prove the entropy monotonicity theorem on a family of flows which interpolate the geodesic flow and the gradient flow on the Wasserstein space over compact Riemannian manifolds. This is based on my paper in Math Ann 2012 and recent joint work with my PhD student Songzi Li.
reads
Xiangdong Li
2013-12-05
11:00:00 - 12:00:00
308 , Mathematics Research Center Building (ori. New Math. Bldg.)
In his seminal paper, Perelman introduced the $W$-entropy for the Ricci flow and proved its monotonicity. This plays a crucial role in the proof of the no local collasping theorem and ”removes the major stumbling block in Hamilton’s approach to geometrization”. In this talk, after a brief review of Perelman's $W$-entropy formula for Ricci flow, I will present our results on the $W$-entropy formula for the heat equation of the Witten Laplacian on manifolds with weighted measure. I will also give a probabilistic interpretation of Perelman's mysterious $W$-entropy for Ricci flow. Then I will present some results on the optimal transport problems for the Fokker-Planck diffusions on manifolds equipped with Perelman's Ricci flow, which can be viewed a natural correspondence of some previous results due to Otto, Lott-Villani, von Renesse-Sturem, McCann-Topping and Lott, etc. We point out that there is an interesting similarity between our $W$-entropy formula for the Witten Laplacian and Lott-Villani and Sturm's result on the monotonicity of the Boltzmann entropy along geodesic on the Wasserstein space over compact Riemannian manifolds. Finally we prove the entropy monotonicity theorem on a family of flows which interpolate the geodesic flow and the gradient flow on the Wasserstein space over compact Riemannian manifolds. This is based on my paper in Math Ann 2012 and recent joint work with my PhD student Songzi Li.