Fall 2024 MA 562
Introduction to Differential Geometry and Topology

Weekly Calendar (back to homepage)

Note: dead links will get filled in as the semester proceeds.

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).

Brief Course Overview

Notes

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.

Notes

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

Notes

8/28. Smooth bump functions and partitions of unity.

Reading: Second half of Chapter 2

Notes

8/30. Tangent space.

Reading: First half of Chapter 3

Notes

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

Notes

9/6. Quiz 1. Tangent bundle.

Reading: Start on Chapter 4.

Study Guide for Quiz 1

Notes

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

Notes

9/11. Rank theorem.

Reading: remainder of Chapter 4 (you don't need to focus too much on the material about covering maps)

Notes

9/13. Embeddings, properties of submersions, embedded submanifolds, level sets.

Reading: first half of Chapter 5

Notes

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

Notes

9/18. Restricting (maps and tangent bundles) to submanifolds.

Reading: end of Chapter 5

Notes

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).

Notes

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.

Notes

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.

Notes

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

Notes

10/2. Pushforwards. Derivations = vector fields. Lie brackets.

Reading: Middle of Chapter 8

Notes

10/4. Lie algebra. Finite dimensionality of Lie(G).

Reading: Finish Chapter 8.

Study Guide for Quiz 2

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

Notes

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:

Notes

10/16. BEGINNING OF PART IV. Vector bundles.

Reading: First two sections of Chapter 10.

Notes

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.

Notes

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.

Notes

10/23. Differentials of functions as covector fields, aka differential 1-forms. Pullbacks of 1-forms.

Reading: Middle of Chapter 11.

Notes

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.

Notes

10/30. Symmetric and alternating tensors.

Reading: Chapter 12

Study Guide for Quiz 3

Finishing notes from Monday

11/1. Exterior algebra I.

Reading: First section of Chapter 14.

Notes

HW11 (submit on Brightspace before the beginning of class on Monday, 11/4)

Week 12

11/4. Exterior algebra II. Differential forms.

Reading: Chapter 14

Notes

11/6. Exterior derivatives.

Reading: Chapter 14.

Notes

11/8. Orientations.

Reading: First two sections of Chapter 15.

Notes

HW12 (submit on Brightspace before the beginning of class on Monday, 11/11)

Week 13

11/11. Integration of differential forms.

Reading: Sections 1-2 of Chapter 16.

Notes

11/13. Stoke's theorem.

Reading: Section 3 of Chapter 16.

Notes

11/15. De Rham cohomology.

Reading: First two sections of Chapter 17.

Notes

HW13 (submit on Brightspace before the beginning of class on Wednesday, 11/20)

Week 14

11/18. Applications of the Mayer-Vietoris sequence.

Reading: Section 3 of Chapter 17.

Notes

11/20. Degree theory. END OF PART IV

Reading: Section 4 of Chapter 17.

Notes

11/22. Brouwer fixed point theorem. BEGINNING OF PART V. Riemannian manifolds

Reading: First two sections of Chapter 13

Notes

HW14 (submit on Brightspace before the beginning of class on Monday, 11/25)

Week 15

11/25. Tangent-cotangent isomorphism. Div grad curl vs exterior derivative. Riemannian volume forms. Divergence theorem.

Reading: Section 3 of Chapter 13. Section 3 of Chapter 15. Section 5 of Chapter 16.

Notes

11/27. Thanksgiving break. No class.

Quiz 4 is a take home quiz. I will send more info on Brightspace. Here is the study guide.

11/29. Thanksgiving break. No class.

HW15 (submit on Brightspace before the beginning of class on Monday, 12/2)

Week 16

12/2. Morse theory.

Reading: Skimming the earlier portions of Milnor's book Morse Theory

Notes

12/4. Hopf index theorem.

Reading: Chapter 6 of Milnor's little book Topology from the Differentiable Viewpoint

Notes

12/6. Gauss-Bonnet theorem.

Reading: Section 4.5 of do Carmo's Differential Geometry of Curves and Surfaces

Notes

No HW (quiet period)

FINAL EXAM (schedule subject to change): Tuesday, December 10, 7:00PM-9:00PM in BRNG 1245