In [13, 14, 15], Cheeger-Colding established the deep structure theory on Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature below. Moreover Jiang and Naber with them in [16, 17] proved further structure results on such spaces. On the other hand Cheeger-Colding asked in an appendix of  whether their theory can be covered by a synthetic way. Now we know the best answer to this question, namely RCD spaces give the best framework in a synthetic treatment of Ricci curvature lower bounds, in order to cover the theory on limit spaces as above. In the three lectures, we will introduce the basics, the techniques, and the recent results for RCD spaces. In particular we will focus on blow-up analysis on such spaces, which play significant roles in many situations. Finally I will provide open problems related to this topic.
No specific advanced knowledge is needed, but it would be helpful to be familiar with the basics of Riemannian geometry.
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Lecture 1.pdf Lecture 2.pdf Lecture 3.pdf