| Monday | Wednesday | Friday | Homework | |
| Week 1 |
8/19. Logistics. Getting to know each other. Course overview. Some point-set topology. Reading: front matter, Appendix A for review, Chapter 1 (pp 1-9). |
8/21. Topological manifolds. Coordinate charts. First examples. Reading: Chapter 1 (pp. 11-29). Feel free to skip Prop 1.16 when reading Chapter 1. |
8/23. Atlases and smooth structures. Smooth manifolds. Real projective space. Reading: finish Chapter 1 Notes continued from Wednesday |
HW1 (submit on Brightspace before the beginning of class on Monday, 8/26) |
| Week 2 |
8/26. Smooth maps. Diffeomorphisms. Examples. Reading: First half of Chapter 2 |
8/28. Smooth bump functions and partitions of unity. Reading: Second half of Chapter 2 |
8/30. Tangent space. Reading: First half of Chapter 3 |
HW2 (submit on Brightspace before the beginning of class on Wednesday, 9/4) |
| Week 3 |
9/2. Labor Day. No class. |
9/4. Derivatives. Reading: Second half of Chapter 3 |
9/6. Quiz 1. Tangent bundle. Reading: Start on Chapter 4. |
HW3 (submit on Brightspace before the beginning of class on Wednesday, 9/11) |
| Week 4 |
9/9. Global derivative. END OF PART I. BEGINNING OF PART II. Immersions, submersions, local diffeomorphisms. Inverse function theorem. Reading: first half of Chapter 4 |
9/11. Rank theorem. Reading: remainder of Chapter 4 (you don't need to focus too much on the material about covering maps) |
9/13. Embeddings, properties of submersions, embedded submanifolds, level sets. Reading: first half of Chapter 5 |
HW4 (submit on Brightspace before the beginning of class on Monday, 9/16) |
| Week 5 |
9/16. More on level sets. Regular values. More on embedded submanifolds. Immersed submanifolds. Reading: middle of Chapter 5 |
9/18. Restricting (maps and tangent bundles) to submanifolds. Reading: end of Chapter 5 |
9/20. Statement of Sard's theorem. Whitney embedding theorem (compact case in class). Reading: first half of Chapter 6. You should make sure you understand the statement of Sard's theorem (although we will not be covering the proof). |
HW5 (submit on Brightspace before the beginning of class on Monday, 9/23) |
| Week 6 |
9/23. Tubular neighborhoods. Transversal intersections. Reading: second half of Chapter 6. The Whitney approximation theorems are very cool, but don't sweat them too much. We will focus on transversality. |
9/25. Transversality. END OF PART II. BEGINNING OF PART III. Lie groups. Reading: Beginning of chapter 7. Notes: look at the ones from Friday of this week. |
9/27. Lie group homomorphisms and Lie subgroups. Reading: Finish Chapter 7. You can skip the subsection on universal covers and the section on equivariant maps for now, although we may need to come back to them at some point later. |
HW6 (submit on Brightspace before the beginning of class on Monday, 9/30) |
| Week 7 |
9/30. Vector fields, frames. Reading: First two sections of Chapter 8 |
10/2. Pushforwards. Derivations = vector fields. Lie brackets. Reading: Middle of Chapter 8 |
10/4. Lie algebra. Finite dimensionality of Lie(G). Reading: Finish Chapter 8. Finishing Wednesday's notes. |
HW7 (submit on Brightspace before the beginning of class on Wednesday, 10/9) |
| Week 8 |
10/7. Fall break. No class. |
10/9. Parallelizability of Lie groups. Induced Lie algebra homomorphisms. Integral curves, global flows. Reading: First two sections of Chapter 9 |
10/11. Flows. Lie derivative. Reading: Sections 3 and 4 of of Chapter 9 Finishing Wednesday's notes. |
HW8 (submit on Brightspace before the beginning of class on Monday, 10/21) |
| Week 9 |
10/14. Lie derivative = Lie bracket. Commuting flows. END OF PART III. Reading: |
10/16. BEGINNING OF PART IV. Vector bundles. Reading: First two sections of Chapter 10. |
10/18. Sections of vector bundles. Trivial if and only if exists a global frame. Bundle homomorphisms. Subbundles. Reading: Sections 3 and 4 of Chapter 10. |
HW9 (submit on Brightspace before the beginning of class on Monday, 10/21) |
| Week 10 |
10/21. Dual spaces and covectors. Cotangent bundle. Reading: First two sections of Chapter 11. |
10/23. Differentials of functions as covector fields, aka differential 1-forms. Pullbacks of 1-forms. Reading: Middle of Chapter 11. |
10/25. Coordinate independent line integrals. Conservative covector fields. Reading: Last two sections of Chapter 11. Continuing notes from Wednesday. |
HW10 (submit on Brightspace before the beginning of class on Monday, 10/28) |
| Week 11 |
10/28. Tensors and tensor fields. Reading: Chapter 12. Feel free to skip the material on Lie derivatives of tensor fields. |
10/30. Symmetric and alternating tensors. Reading: First section of Chapter 14. |
11/1. Differential forms and exterior derivatives. Reading: Last two sections of Chapter 14. |
HW11 (submit on Brightspace before the beginning of class on Monday, 11/4) |
| Week 12 |
11/4. Orientations. Reading: First two sections of Chapter 15. |
11/6. Integration of differential forms. Reading: First two sections of Chapter 16. |
11/8. Stokes's Theorem. Reading: Section 3 of Chapter 16. |
HW12 (submit on Brightspace before the beginning of class on Monday, 11/11) |
| Week 13 |
11/11. De Rham cohomology. Reading: Sections 1-3 of Chapter 17 |
11/13. Degree theory. Brouwer fixed point theorem. END OF PART IV. Reading: Section 4 of Chapter 17. |
11/15. BEGINNING OF PART V. Riemannian manifolds. Reading: First two sections of Chapter 13. |
HW13 (submit on Brightspace before the beginning of class on Monday, 11/18) |
| Week 14 |
11/18. More on Riemannian manifolds. Reading: |
11/20. Morse functions and Morse homology. Reading: |
11/22. Existence of Morse functions on smooth manifolds. Reading: |
HW14 (submit on Brightspace before the beginning of class on Monday, 11/25) |
| Week 15 |
11/25. Hopf index theorem. Reading: |
11/27. Thanksgiving break. No class. |
11/29. Thanksgiving break. No class. |
HW15 (submit on Brightspace before the beginning of class on Monday, 12/2) |
| Week 16 |
12/2. Gauss-Bonnet theorem. Reading: |
12/4. Reading: |
12/6. Reading: |
No HW (quiet period) |
| FINAL EXAM (schedule subject to change): | Tuesday, December 10, 7:00PM-9:00PM in BRNG 1245 |