## Analysis and Geometry of Metric Measure Spaces: Lecture Notes of the 50th Seminaire De Mathematiques Superieures (SMS), Montreal, 2011This book contains lecture notes from most of the courses presented at the 50th anniversary edition of the Seminaire de Mathematiques Superieures in Montreal. This 2011 summer school was devoted to the analysis and geometry of metric measure spaces, and featured much interplay between this subject and the emergent topic of optimal transportation. In recent decades, metric measure spaces have emerged as a fruitful source of mathematical questions in their own right, and as indispensable tools for addressing classical problems in geometry, topology, dynamical systems, and partial differential equations. The summer school was designed to lead young scientists to the research frontier concerning the analysis and geometry of metric measure spaces, by exposing them to a series of minicourses featuring leading researchers who highlighted both the state-of-the-art and some of the exciting challenges which remain. This volume attempts to capture the excitement of the summer school itself, presenting the reader with glimpses into this active area of research and its connections with other branches of contemporary mathematics. |

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### Contents

Analysis on the Sierpinski Carpet | 27 |

Heat Kernel Estimates SobolevType Inequalities and Riesz Transform | 55 |

Lectures on MaTrudingerWang Curvature and Regularity of Optimal | 119 |

Geometry Regularity | 142 |

A Proof of Bobkovs Spectral Bound For Convex Domains via Gaussian Fitting | 181 |

A Visual Introduction to Riemannian Curvatures and Some Discrete | 197 |

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Alexandrov Amer Anal argument assume ball Borel boundary c-convex CD(K coarse Ricci curvature compact cone of type constant converge convex function cost function curve deﬁne Deﬁnition denote density dimension Dirichlet form distance domains entropy equivalent estimates Euclidean example exists Figalli ﬁrst ﬂow fractals Gaussian geodesic geometry gradient flow graph Harnack inequality Hd(E heat equation heat kernel Hölder Hölder continuity holds implies isoperimetric Kantorovich Lecture Notes Lemma Lipschitz Loeper log-concave lower bound Math metric measure spaces metric spaces minimal cone minimal sets Monge–Ampère Monge–Ampère equation optimal maps optimal transport maps parallel transport Partial Differential Equations Plateau problem point of type probability measures proof prove R. J. McCann regularity of optimal regularity results Ricci curvature Riemannian manifolds round sphere satisﬁes sectional curvature Sierpinski carpet smooth soap films solutions tangent vector Theorem 3.4 theory