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Title page

Chapter 1. Introduction

1.1. The simple case of normed spaces

1.2. The general situation

Chapter 2. Preliminaries

2.1. Metric spaces and Wasserstein distance

2.2. Metric measure spaces

2.3. Sobolev classes

2.4. The Cheeger energy and its gradient flow

Chapter 3. Differentials and gradients

3.1. Definition and basic properties

3.2. Horizontal and vertical derivatives

3.3. Calculus rules

Chapter 4. Laplacian

4.1. Definition and basic properties

4.2. Calculus rules

4.3. The linear case

Chapter 5. Comparison estimates

5.1. Weak Ricci curvature bounds

5.2. Variants of calculus rules

5.3. Proof of Laplacian comparison

Appendix A. On the duality between cotangent and tangent spaces

Appendix B. Remarks about the definition of the Sobolev classes

References

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