AIS - The Laplacian in Riemannian Geometry (2024)
Speakers and Syllabus
Syllabus:
| Name of the Speaker with affiliation | No. of Lecture s | Detailed Syllabus |
| Outstation speakers | ||
| Dr. Soma Maity, IISER Mohali (SM) | 4 (4x 1.5 hrs) =6hrs |
Eigenvalues and eigenfunctions of the Laplacian in a Riemannian manifold, Sturm Liouville decomposition of the eigenspace of the Laplacian. |
| Dr. Harish Seshadri, IISc Bangalore (HS) | 4 (4x 1.5 hrs) =6hrs |
First eigenvalues of Laplacian and its relations to geodesics and curvatures, Laplacian in space forms, comparison theorems for sectional curvature bounded above and Ricci curvature bounded below. |
| Dr. Kingshook Biswas, ISI Kolkata (KB) | 4 (4x 1.5 hrs) =6hrs |
First eigenvalues of Laplacian : the co-area formula, the Faber Krahn inequality, the Cheeger, Sobolev and isoperimetric constants, the Sobolev constants and eigenvalue, eigenfunction estimates. |
| Local speakers | ||
| Sombuddha Bhattacharyya (SB) | 4 (4x 1.5 hrs) =6hrs |
Laplacian on domains in the Euclidean space, The spectral theorem for the Laplace operator with Dirichlet boundary condition and Neumann boundary condition on a bounded domain in \mathbb{R}^n. (will need Lax-Milgram theorem, and Rellich’s lemma) |
| Dr. Atreyee Bhattacharya (AB) | 4 (4x 1.5 hrs) =6hrs |
Introduction to Riemannian Geometry: Riemannian metric, and Riemannian connections, parallel transport, geodesics, exponential map, convex neighborhoods, geodesics as locally minimizing curves, completeness of Riemannian metrics, the Hopf-Rinow Theorem. |
| Dr. Anandateertha Mangasuli (AM) | 4 (4x 1.5 hrs) =6hrs |
Riemann Curvature Tensor, Bianchi identity, sectional, Ricci and scalar curvatures, properties of curvature, Lie and covariant derivatives of tensors, gradient and Hessian of smooth functions, divergence of vector fields, rough Laplacian on tensors of a Riemannian manifold. |
References:
- Evans, Partial Differential equations, AMS, (2010)
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer (2004).
- I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press,Inc. (1984)
Tutorial Assistants:
| S. No. | Name | Affiliation |
| 1 | Ms Sayoojya Prakash (SP) | PhD student, IISER Bhopal |
| 2 | Mr Satyajit Maity (SaM) | PhD student, IISER Bhopal |
| 3 | Ms Jahnabi Chakraborty (JC) | PhD student, IISER Bhopal |
Time Table
| Day | Date | Lecture 1 (9.30–11. 00) |
Tea (11.05 to 11.25) |
Lecture 2 (11.30–1.00) |
Lunch (1.05-2.25) |
Tutorial (2.30–3.30) |
Tea (3.35-3.55) |
Tutorial (4.00-5.0 0) |
Snacks 5.05 to 5. 30 |
| (name of the speaker | (name of the speaker | (name of the speaker + tutors) | (name of the speaker + tutors) | ||||||
| Mon | 02/12/2024 | SB | AB | SB, SaM | AB, SP | ||||
| Tues | 03/12/2024 | SB | AB | SB, SaM | AB, SP | ||||
| Wed | 04/12/2024 | SB | AM | SB, SaM | AM, JC | ||||
| Thu | 05/12/2024 | AB | AM | AB, SP | AM, JC | ||||
| Fri | 06/12/2023 | AB | AM | AB, SP | AM, JC | ||||
| Sat | 07/12/2024 | SB | AM | SB, SaM | AM, JC | ||||
| SUNDAY:OFF | |||||||||
| Mon | 09/12/2024 | SM | HS | SM, SP | HS, JC | ||||
| Tues | 10/12/2024 | SM | HS | SM, SP | HS, JC | ||||
| Wed | 11/12/2024 | KB | HS | KB, SaM | HS, JC | ||||
| Thu | 12/12/2024 | KB | HS | KB, SaM | HS, JC | ||||
| Fri | 13/12/2024 | SM | KB | SM, SP | KB, SaM | ||||
| Sat | 14/12/2024 | SM | KB | SM, SP | KB, SaM | ||||