The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

## Further Information

**Aimed at: **people interested on Geometric Analysis, Geometric Measure Theory and regularity theory in Partial Differential Equations.

**Prerequisites: **Riemannian and Differential Geometry, Metric spaces, basic knowledge of Partial Differential Equations.

**Outline of the course:**

**Lecture 1:**- Quick introduction to non-smooth spaces with lower Ricci curvature bounds [1, 23, 20, 17];
- Basic properties of spaces with lower Ricci bounds: Bishop-Gromov inequality and doubling metric measure spaces, Bochner’s inequality, splitting theorem [19, 22];
- Convergence and stability: Gromov-Hausdorff convergence, Gromov pre-compactness theorem, stability and tangent cones [19, 22];

__Lecture 2:__- Functional form of the splitting theorem via splitting maps;
- δ-splitting maps and almost splitting theorem [5, 7];
- Definition of metric measure cone, stability of RCD property for cones [16];
- Functional form of the volume cone implies metric cone [12];
- Almost volume cone implies almost metric cone via stability.

__Lecture 3:__- Maximal function type arguments;
- Existence of Euclidean tangents;
- Rectifiability and uniqueness of tangents at regular points [18];
- Volume convergence [9, 13];
- Tangent cones are metric cones on noncollapsed spaces [5, 6, 13].

__Lecture 4:__- Euclidean volume rigidity [9, 6, 13];
- ε-regularity and classical Reifenberg theorem [6, 15, 7];
- Harmonic functions on metric measure cones, frequency and separation of variables [7];
- Transformation theorem for splitting maps [7];
- Proof of canonical Reifenberg theorem via harmonic splitting maps [7].

__Lecture 5:__- Regular and singular sets [6, 13];
- Stratification of singular sets [6, 13];
- Examples of singular behaviours [10, 11];
- Hausdorff dimension bounds via Federer’s dimension reduction [6, 13];
- Quantitative stratification of singular sets [8];
- An introduction to quantitative differentiation [3];
- Cone splitting principle [8];
- Quantitative singular sets and Minkowski content bounds [8].

__Lecture 6:__- The aim of this lecture is to give an introduction to the most recent developments of the regularity theory for non collapsed Ricci limit spaces. We will introduce the notion of neck region in this context and then outline how they have been used to prove rectifiability of singular sets in any codimension for non collapsed Ricci limit spaces by Cheeger-Jiang-Naber [7].