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Synthetic theory of Ricci curvature - when information theory, optimization, geometry and gradient flows meet
Published on Aug 20, 20153876 Views
This is the story of a mathematical theory which was born from the encounter of several fields. Those fields were: Riemannian geometry, gradient flows, information, and optimal transport; and their
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Chapter list
Synthetic Theory of Ricci Curvature00:00
Analytic vs. Synthetic: an example00:42
Geometric meaning of curvature10:43
Metric spaces of nonnegative sectional curvature10:56
Examples12:44
Theory of Alexandrov spaces of positive curvature14:00
The meaning of Ricci curvature, I - 115:26
The meaning of Ricci curvature, I - 215:31
The meaning of Ricci curvature, I - 316:36
The meaning of Ricci curvature, I - 417:42
The meaning of Ricci curvature, II19:26
The meaning of Ricci curvature, III - 120:59
The meaning of Ricci curvature, III - 222:04
Consequences of Ricci curvature lower bounds22:50
Example: curved Brunn–Minkowski for Ric ≥ 023:41
Generalizations25:50
Theory of CD(K,N) bounds - 127:16
Theory of CD(K,N) bounds - 227:44
Gradient flows, Boltzmann Entropy and Optimal Transport29:12
Gradient flows in a nonsmooth setting30:46
Information theory34:00
Microscopic meaning of the entropy functional - 135:37
Microscopic meaning of the entropy functional - 236:30
Famous computation by Boltzmann - 136:45
Famous computation by Boltzmann - 237:22
Famous computation by Boltzmann - 337:44
Famous computation by Boltzmann - 437:59
Famous computation by Boltzmann - 538:09
Recall: Sanov’s Theorem38:32
Entropy appears in ...39:40
Optimal transport40:46
The Kantorovich problem - 141:32
The Kantorovich problem - 242:43
Engineer’s interpretation42:58
Geometric structure44:41
Optimal transport appears in ...45:58
Unexpected encounter of the third type46:29
The Lazy Gas experiment - 148:02
The Lazy Gas experiment - 248:06
Relation between transport and Ricci49:37
Picture49:58
Metric-measure spaces of nonnegative Ricci curvature (Lott–Sturm–Villani) - 150:46
Metric-measure spaces of nonnegative Ricci curvature (Lott–Sturm–Villani) - 252:08
General CD (K,N)52:10
Stability52:52
Compatibility (Petrunin 2009)53:32
Properties derived from the synthetic formulation54:35
Isoperimetric inequalities, concentration - 154:45
Isoperimetric inequalities, concentration - 254:46
Rough heat flow (Ambrosio–Gigli–Savar´e 2011)54:47
Side PDE remark55:07
How wide is this generalization?55:32
RCD(K,N) Spaces / RCD∗(K,N) Spaces56:27
Recent success58:06
Adaptation to discrete spaces58:37