Introduction 1
1．Formulation of the optimal transportation problem 1
2．Basic questions 6
3．Overview of the course 10
Chapter 1．The Kantorovich Duality 17
1.1．General duality 17
1.2．Distance cost functions 34
1.3．Appendix:A duality argument in Cb（X×Y） 39
1.4．Appendix:｛0,1｝-valued costs and Strassen's theorem 44
Chapter 2． Geometry of Optimal Transportation 49
2.1．A duality-based proof for the quadratic cost 50
2.2．The real line 75
2.3．Alternative arguments 80
2.4．Generalizations to other costs 87
2.5．More on c-concave functions 105
Chapter 3．Brenier's Polar Factorization Theorem 109
3.1．Rearrangements and polar factorization 109
3.2．Historical motivations:fluid mechanics 113
3.3．Proof of Brenier's polar factorization theorem 121
3.4．Related facts 124
Chapter 4．The Monge-Ampère Equation 127
4.1．Informal presentation 127
4.2．Regularity 133
4.3．Open problems 143
Chapter 5．Displacement Interpolation and Displacement Convexity 145
5.1．Displacement interpolation 145
5.2．Displacement convexity 152
5.3．Application:uniqueness of ground state 166
5.4．The Eulerian point of view 168
Chapter 6．Geometric and Gaussian Inequalities 187
6.1．Brunn-Minkowski and Prékopa-Leindler inequalities 188
6.2．The Alesker-Dar-Milman diffeomorphism 194
6.3．Gaussian inequalities 197
6.4．Sobolev inequalities 204
Chapter 7．The Metric Side of Optimal Transportation 209
7.1．Monge-Kantorovich distances 211
7.2．Topological properties 216
7.3．The real line 222
7.4．Behavior under rescaled convolution 224
7.5．An application to the Boltzmann equation 227
7.6．Linearization 237
Chapter 8．A Differential Point of View on Optimal Transportation 241
8.1．A differential formulation of optimal transportation 242
8.2．Differential calculus in（P（Rn），W2） 254
8.3．Monge-Kantorovich induced dynamics 255
8.4．Time-discretization 260
8.5．Differentiability of the quadratic Wasserstein distance 266
8.6．Non-quadratic costs 270
Chapter 9．Entropy Production and Transportation Inequalities 271
9.1．More on optimal-transportation induced dissipative equations 272
9.2．Logarithmic Sobolev inequalities 283
9.3．Talagrand inequalities 295
9.4．HWI inequalities 301
9.5．Nonlinear generalizations:internal energy 305
9.6．Nonlinear generalizations:interaction energy 308
Chapter 10．Problems 311
List of Problems 312
Bibliography 353
Table of Short Statements 369
Index 375