**Description**

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 [13] 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.

**Prerequisite**

No specific advanced knowledge is needed, but it would be helpful to be familiar with the basics of Riemannian geometry.

**References**

[1] L. Ambrosio, Calculus, heat flow and curvature-dimension bounds in metric measure spaces, Proceedings of the ICM 2018, Vol. 1, World Scientific, Singapore, (2019), 301–340.

[2] L. Ambrosio, N. Gigli, G. Savare, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math. 195 (2014), 289--391.

[3] L. Ambrosio, N. Gigli, G. Savare, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), 1405--1490.

[4] L. Ambrosio, N. Gigli, G. Savare, Bakry-\'Emery curvature-dimension condition and Riemannian Ricci curvature bounds.Ann. of Prob. 43 (2015), 339--404.

[5] L. Ambrosio, S. Honda, New stability results for sequences of metric measure spaces with uniform Ricci bounds from below, Measure theory in non-smooth spaces, 1–51, Partial Differ. Equ. Meas. Theory,De Gruyter Open, Warsaw, 2017.

[6] L. Ambrosio, S. Honda, Local spectral convergence in RCD^*(K, N) spaces. Nonlinear Anal. 177 Part A (2018), 1–23.

[7] L. Ambrosio, D. Trevisan, Well-posedness of Lagrangian flows and continuity equations in metric measure spaces, Anal. PDEs. 7 (2014), 1179–-1234.

[8] G. Antonelli, E. Brue, D. Semola, Volume Bounds for the Quantitative Singular Strata of Non Collapsed RCD Metric Measure Spaces, Anal. Geom. Metr. Spaces 7 (2019), no. 1, 158-178.

[9] C. Brena, N. Gigli, S. Honda, X. Zhu, Weakly noncollapsed RCD spaces are strongly noncollapsed, arXiv:2110.02420

[10] E. Brue, E. Pasqualetto, D. Semola, Rectifiability of RCD(K,N) spaces via \delta-splitting maps, Ann. Fenn. Math. 46 (2021), no. 1,465-482.

[11] E. Brue, D. Semola, Constancy of the dimension for RCD(K, N)spaces via regularity of Lagrangian flows. Comm. Pure Appl. Math. 73 (2020), 1141--1204.

[12] F. Cavalletti, E. Milman, The Globalization Theorem for the Curvature Dimension Condition, Invent. Math. 226 (2021), no. 1, 1–137

[13] J. Cheeger, T. H. Colding, On the structure of spaces with Ricci curvature bounded below, I. J. Differential Geom. 46 (1997), 406--480.

[14] J. Cheeger, T. H. Colding, On the structure of spaces with Ricci curvature bounded below, II. J. Differential Geom. 54 (2000), 13--35.

[15] J. Cheeger, T. H. Colding, On the structure of spaces with Ricci curvature bounded below, III. J. Differential Geom. 54 (2000), 37--74.

[16] J. Cheeger, W. Jiang, A. Naber, Rectifiability of singular sets of non collapsed limit spaces with Ricci curvature bounded below, Ann. of Math. 193 (2021), 407-538.

[17] T. H. Colding, A. Naber, Sharp H\"older continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. of Math. 176 (2012), 1173--1229.

[18] Q. Deng, H\"older continuity of tangent cones in RCD(K,N) spaces and applications to non-branching, arXiv:2009.07956.

[19] G. De Philippis, N. Gigli, From volume cone to metric cone in the nonsmooth setting. Geom. Funct. Anal. 26 (2016), no. 6, 1526–1587

[20] G. De Philippis, N. Gigli, Non-collapsed spaces with Ricci curvature bounded from below. J. Ec. polytech. Math. 5 (2018),613–650.

[21] M. Erbar, K. Kuwada, K.-T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces, Invent. Math. 201 (2015), 993--1071.

[22] N. Gigli, The splitting theorem in non-smooth context, arXiv:1302.5555.

[23] N. Gigli, Nonsmooth differential geometry --An approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math.Soc. 251 (2018), no. 1196.

[24] N. Gigli, A. Mondino, G. Savare,Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows. Proc. Lond. Math. Soc. (3), 111 (2015),1071-1129.

[25] S. Honda, New differential operator and noncollapsed RCD spaces, Geom. Topol. 24 (2020), 2127-2148.

[26] S. Honda, Y. Peng, A note on the topological stability theorem from RCD spaces to Riemannian manifolds, arXiv:2202.06500

[27] A. Mondino, A. Naber, Structure theory of metric measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc. 21 (2019),1809–-1854.

[28] J. Pan, G. Wei, Examples of Ricci limit spaces with non-integer Hausdorff dimension, to appear in Geom. Funct. Anal.

[29] K.-T. Sturm, On the geometry of metric measure spaces, I, Acta Math.196 (2006), 65--131.

[30] K.-T. Sturm, On the geometry of metric measure spaces, II, Acta Math. 196 (2006), 133--177.

[31] B. Wang, X. Zhao, Canonical diffeomorphisms of manifolds near spheres, arXiv:2109.14803.

**Notes:**

Lecture 1.pdf Lecture 2.pdf Lecture 3.pdf