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MATH 6230 - Differential Geometry - Spring 2017
Class Schedule
This schedule remains tenative, and is subject to change. All reading will be from the main textbook for the course (Lee, Introduction to Smooth Manifolds, 2nd Edition) unless specified otherwise.
| Week | Dates | Topics | Reading | Notes |
|---|---|---|---|---|
| 0 | You should be comfortable with the material in Appendices B and C (at least the statements), and it would not hurt to glance at Appendices A and D. We will be going over the implicit function theorem and (probably) the existence and uniqueness of solutions of ordinary differential equations in the course, and we will grab what we need from Appendix A as we need it. | |||
| 1 | 01/17-20 | topological and smooth manifolds | Appendix A (especially 596-611), Chapter 1 (especially pages 1-24) | |
| 2 | 01/23-27 | implicit function theorem | Appendix C (especially pages 657-662) | |
| 3 | 01/30-02/03 | the category of smooth manifolds, classifications in very low dimensions | Chapter 2, pages 32-42.5. | Homework 01 Due: Wednesday Feb 1 |
| 4 | 02/06-10 | tangent vectors and tangent bundles | Chapter 3 (especially pages 50-71) | |
| 5 | 02/13-17 | the Rank Theorem | Chapter 4 | Homework 02 Due: Friday Feb 17 |
| 6 | 02/20-24 | submanifolds | Chapter 5 (especially pages 98-120) | |
| 7 | 02/27-03/03 | Sard's Theorem, Whitney's Embedding Theorem, Transversality, submersions, Lie groups, Quotient Manifold Theorem | Chapter 6 (especially pages 125-136, the tubular neighbourhood theorem part 137-141, and 143-147) and Chapter 7 (especially pages 150-154, 156-168), and pages 544-547 (the quotient manifold theorem) of the text. | Homework 03 Due: Wednesday Mar 1 |
| 8 | 03/06-10 | vector fields | Chapter 8 | |
| 9 | 03/13-17 | integral curves and flows, Lie groupoids | Appendix D, Chapter 9 (especially pages 205-217), Notes on Lie Groupoids | Homework 04 Due: Wednesday Mar 15 |
| 10 | 03/20-24 | vector bundles and K-Theory, multi-linear algebra, tensors, tensor fields, and Lie derivatives | Chapter 10 (pages 249-264), Notes on K-Theory, Chapter 9 (pages 217-220, 227-236), Chapters 11 (pages 272-278, 280-287), Chapter 12, and Chapter 14 (pages 350-358) | |
| Spring Break! - no classes | ||||
| 11 | 04/03-07 | Riemannian manifolds | Chapter 13 | Homework 05 Due: Wednesday Apr 5 |
| 12 | 04/10-14 | differential forms and orientation | Chapter 14, Chapter 15 (especially pages 377-386) | |
| 13 | 04/17-21 | integration and Stokes' Theorem | Chapter 16 (especially pages 400-415) | Homework 06 Due: Wednesday Apr 19 |
| 14 | 04/24-28 | de Rham cohomology | Chapter 17 | |
| 15 | 05/01-05 | connections and curvature | Various parts of Lee's Riemannian Manifolds: an Introduction to Curvature | Homework 07 Due: Wednesday May 3 |