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Synthetic theory of ricci curvature bounds
Cédric Villani
- Vol. 15, pp 81-121
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Synthetic theory of Ricci curvature bounds is reviewed in this article, from the conditions which led to its birth, up to some of its latest developments, with a review of the most recent developments.Abstract:
Synthetic theory of Ricci curvature bounds is reviewed, from the conditions which led to its birth, up to some of its latest developmentsread more
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Non-collapsed spaces with Ricci curvature bounded from below
Guido De Philippis,Nicola Gigli +1 more
TL;DR: In this paper, a non-collapsed space with Ricci curvature bounded from below is defined, and the versions of Colding's volume convergence theorem and of Cheeger-Colding dimension gap estimate are proved.
Journal ArticleDOI
Infinitesimal Hilbertianity of weighted Riemannian manifolds
Danka Lučić,Enrico Pasqualetto +1 more
TL;DR: The main result of as mentioned in this paper is that any weighted Riemannian manifold is Hilbertian and that all weighted (sub-Riemannians) Carnot groups are infinitesimally Hilbertian.
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Rigidity properties of the hypercube via Bakry-Emery curvature
TL;DR: The first known discrete analogues of Cheng's and Obata's rigidity theorems for the discrete Bonnet-Myers diameter bound and the Lichnerowicz eigenvalue estimate were given in this article.
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The Serre–Swan theorem for normed modules
Danka Lučić,Enrico Pasqualetto +1 more
TL;DR: In this paper, the authors analyse the structure of the $$L^0$$¯¯¯¯ -normed $$L€ 0$$¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ -modules over a metric measure space. But they do not discuss under which conditions an µ-normed µ-module can be viewed as the space of sections of a suitable measurable Banach bundle and in which sense such correspondence can be actually made into an equivalence of categories.
References
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Book
Optimal Transport: Old and New
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Book
Topics in Optimal Transportation
TL;DR: In this paper, the metric side of optimal transportation is considered from a differential point of view on optimal transportation, and the Kantorovich duality of the optimal transportation problem is investigated.
Book
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.
Book
A Course in Metric Geometry
TL;DR: In this article, a large-scale Geometry Spaces of Curvature Bounded Above Spaces of Bounded Curvatures Bounded Below Bibliography Index is presented. But it is based on the Riemannian metric space.
Book
The concentration of measure phenomenon
TL;DR: Concentration functions and inequalities isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities Sharp bounds of Gaussian and empirical processes Selected applications References Index