Abstract: Ricci limit spaces appear naturally as Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded below. These spaces were discovered and extensively studied by J.Cheeger and T.Colding inspiring several innovative developments in Riemannian Geometry and in different synthetic approaches to lower curvature bounds such as Alexandrov Geometry and the theory of CD spaces. This series of four lectures is intended as a detailed introduction to the subject covering both the main ideas as well as important technical details in the theory.
Online lectures. Zoom link (Meeting ID: 912 9010 6599; Passcode: 117902)
First lecture: November 16th, 2021, 9:00am Mexico / 4:00pm Italy.
• Video • Lecture Notes: 1.1 | 1.2 |1.3Topics
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Second Lecture: November 23th, 2021, 9:00am Mexico / 4:00pm Italy
• Video • Lecture Notes: 2.1 | 2.2Topics
- The smooth splitting theorem (statement and sketch of the proof);
- The smooth almost splitting theorem (statement);
- Splitting theorem for Ricci-limit spaces (statement and sketch of the proof);
- Discussion of some applications and related results in the smooth and nonsmooth setting: estimates on the Hausdorff dimension of singular strata of non collapsed Ricci-limit spaces (ncRLS), volume estimates of the quantitative singular strata of ncRLS, rectifiability of singular strata of ncRLS, boundary of ncRLS (statements);
- Splitting theorem in the nonsmooth setting (statement);
- $\delta$-Splitting maps: from \delta-splitting maps to splitting spaces and vice-versa in the smooth and nonsmooth setting (statements);
- Under Ricci lower bounds, GH-close to Euclidean balls if and only if Volume-close to Euclidean balls [in the smooth and nonsmooth setting] (statement);
- Reifenberg and Volume convergence [in the smooth and nonsmooth setting] (statement);
- First part of the proof of the smooth almost splitting theorem: from splitting maps to splitting spaces
Third Lecture: November 30th, 2021, 9:00am Mexico / 4:00pm Italy
• Video • Lecture Notes: 3.1 | 3.2Topics
- Final part of the proof of the Proposition "from splitting maps to splitting spaces";
- Conclusion of the proof of the almost splitting theorem: construction of a $(1,\delta)$-splitting map;
- From almost volume cones to almost metric cones in the smooth setting (statement);
- Definition of a metric cone;
- Definition of $(K,N)$-metric measure cones over metric measure spaces;
- Stability of the bounds from below on the curvature by means of the cone constructions (statement).
Fourth Lecture (final): January 11, 2022, 9:00am Mexico / 4:00pm Italy
• Video • Lecture Notes: 4.1 | 4.2Topics
- From volume cones to metric cones in the nonsmooth setting (statement);
- A corollary: the tangent spaces of a non-collapsed Ricci limit spaces are metric cones [in the smooth and non-smooth setting] (statement);
- Proof of the "almost volume cone implies almost metric cone" theorem in the smooth setting, step I: construction of a $\delta$-conical function;
- Proof of the "almost volume cone implies almost metric cone" theorem in the smooth setting, step II: from the $\delta$-conical function to the conclusion of the theorem.
Contact:
- Gil Bor (CIMAT) gil@cimat.mx
- Oscar Palmas Velasco (UNAM) oscar.palmas@ciencias.unam.mx
- Jesús Núñez Zimbrón (CIMAT) jesús.nunez@cimat.mx