Class: TuTh, 8:30-10:00AM, Online on Microsoft teams. If you are interested in attending, please send an email to me.
Instructor: Ved Datar
Email: vv lastname at math.iisc.ac.in, no spaces
Office: X05
Office hours: By appointment
Reference books: Peter Petersen, Riemannian geometry , Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998.
Richard Schoen and ST Yau, Lectures of Differential Geometry, International Press, 1997.
Jeff Cheeger, Degenerations of Riemannian metrics under Ricci curvature bounds, Publications of the Scuola Normale Superiore, Birkhauser, 2001.Pre-requisites: MA 333 - Riemannian Geometry
Topics to be covered: Bochner formula, Laplace comparison, Volume comparison, Heat kernel estimates, Cheng-Yau gradient estimates, Cheeger-Gromoll splitting theorem, Gromov-Haudorff convergence, epsilon regularity, almost rigidity, quantitative structure theory of Riemannian manifolds with Ricci curvature bounds. If time permits, we will discuss the proof of the co-dimension four conjecture due to Cheeger and Naber.
Homeworks - 50%, Final presentation - 50%
There will be 6 homeworks. The best five will be counted towards the grade. There is no late submission of homeworks.
To pass the class, you have to make the final presentation. The presentation will be of half an hour on on a topic chosen from a list of topics that shall be announced later
| Number | Date | Topic | Homework | Notes |
| 1 | Th 10/01 | An overview of the course | ||
| 2 | Tu 10/06 | Review of Riemannian geometry | ||
| 3 | Th 10/08 | Bochner formula, Local Laplace comparison | ||
| 4 | Tu 10/13 | Volume comparison | ||
| 5 | Th 10/15 | Weak notions of Laplacian, Maximum principle | ||
| 6 | Tu 10/20 | Global Laplace comparison | ||
| 7 | Th 10/22 | Splitting theorem | ||
| 8 | Tu 10/27 | Segment inequalities | ||
| 9 | Th 10/29 | Poincare inequality | ||
| 10 | Tu 11/03 | Gradient estimates for the heat kernel | ||
| 11 | Th 11/05 | Gaussian estimates, harnack inequality |