# Ricci curvature and measures

@article{Bourguignon2009RicciCA, title={Ricci curvature and measures}, author={Jean Pierre Bourguignon}, journal={Japanese Journal of Mathematics}, year={2009}, volume={4}, pages={27-45} }

Abstract.In the last thirty years three a priori very different fields of mathematics, optimal transport theory, Riemannian geometry and probability theory, have come together in a remarkable way, leading to a very substantial improvement of our understanding of what may look like a very specific question, namely the analysis of spaces whose Ricci curvature admits a lower bound. The purpose of these lectures is, starting from the classical context, to present the basics of the three fields that…

## 5 Citations

### The Calabi metric for the space of Kähler metrics

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Given any closed Kähler manifold we define, following an idea by Calabi (Bull. Am. Math. Soc. 60:167–168, 1954), a Riemannian metric on the space of Kähler metrics regarded as an infinite dimensional…

### A glimpse into the difierential topology and geometry of optimal transport

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This note exposes the difierential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures:…

### The Monge-Kantorovich problem: achievements, connections, and perspectives

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This article gives a survey of recent research related to the Monge-Kantorovich problem. Principle results are presented on the existence of solutions and their properties both in the Monge optimal…

### An Elementary Introduction to Information Geometry

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The fundamental differential-geometric structures of information manifolds are described, the fundamental theorem of information geometry is state, and some use cases of these information manifolding in information sciences are illustrated.

### Geometric Quantization of Complex Monge-Ampère Operator for Certain Diffusion Flows

- MathematicsGSI
- 2013

In the 40’s, C.R. Rao considered probability distributions for a statistical model as the points of a Riemannian smooth manifold, where the considered RiemANNian metric is the so-called Fisher metric, which is actually the Fubini-Study metric.

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