- Course Description -

This course is a graduate-level introduction to smooth manifolds and calculus on manifolds. This course is primarily aimed at preparing PhD students in mathematics for the screening exams and for math research.

Topics will include vector bundles, special maps and sub-manifolds, tensors, differential forms and de Rham theory.

We will assume a background in undergraduate real analysis, point-set topology and basic abstract algebra.

- Course Info -

- Instructor: Julian Chaidez (julian.chaidez@usc.edu)
- Assistant: Wenhan Jiang (wenhanji@usc.edu)
- Textbook:
*Introduction to Smooth Manifolds.*J. M. Lee - Lectures: MWF 11-12a in KAP 134
- Office Hours: F 1-3 in KAP 245
- Grade Scheme: The course grade will be 60% homework, 20% midterm and 20% final.
- Exam Structure: Each exam will consist of a take home and in class component, weighted equally.
- In-Class Exam Dates: 2/28 (Midterm) and 4/26 (Final)
- For more information, see the course syllabus.

- Homework Schedule -

- All homework is assigned from Lee.
- HW 1 (due 1/22) - Ch 1: 1,2,9 and Ch 2: 3a, 9, 10, 14.
- HW 2 (due 1/29) - Ch 10: 1, 12 and Ch 3: 1, 3, 4.
- HW 3 (due 2/5) - Ch 11: 5, 6, 7.Ch 4: 2,4. Ch 5: 1,2,6.
- HW 4 (due 2/12) - Ch 7: 1,2,5,14/15. Ch 8: 10.
- HW 5 (due 2/21) - Ch 8:23,29. Ch 9:4,5. Ch 12:11,12
- Midterm - Take home component due 2/28.
- HW 6 (due 3/18) - Ch 14: 1, 5, 6, 7. Ch 15: 1, 3, 5.
- HW 7 (due 3/25) - Ch 16: 1, 2, 3, 5, 6, 9.
- HW 8 (due 4/1) - Ch 16: 18, 21, 22. Ch 17: 1,2.
- HW 9 (due 4/8) - Ch 17: 4, 6, 7, 11, 12.
- HW 10 (due 4/22) - Ch 18: 1, 2, 6, 7, 8, 9.
- Final - Take home component due 4/26.

- Lecture Schedule and Readings -

- M 1/8. introduction, topological manifolds. 1-10, 25-27.
- W 1/10. smooth manifolds (with boundary). 11-24.
- F 1/12. smooth functions and maps, partitions of unity. 32-48.
- W 1/17. examples: implicit function theorem, products. 1-24.
- F 1/19. vector bundles, sections. 249-261.
- M 1/22. bundle homomorphisms, pullback, sub-bundles. 261-268.
- W 1/24. tangent vectors, tangent bundle, tangent map. 50-60
- F 1/26. cotangent vectors, cotangent bundle, differentials. 272-287.
- M 1/29. maps of constant rank, immersion, submersions. 77-91
- W 1/31. embeddings, sub-manifolds, normal bundles. 98-108
- F 2/2. Lie groups, examples, homomorphisms. 150-156.
- M 2/5. Lie subgroups, Lie group actions, equivariance. 156-171.
- F 2/9. flows, vector-fields, integral curves. 205-217.
- M 2/12. Lie bracket, Lie derivative. 175-189, 227-231
- W 2/14. Lie algebras. 190-199.
- F 2/16. multilinear algebra, tensors, tensor-fields. 304-313.
- W 2/21. symmetric and antisymmetric tensor-fields. 313-316.
- W 2/21-W 2/26. topics in Riemannian geometry.
- F 3/1. exterior algebra. 349-359.
- M 3/4. differential forms, exterior derivative. 359-367.
- W 3/6. Cartan's magic formula. 369-373.
- F 3/8. orientations, volume forms. 377-388.
- M 3/18. geometry of volume forms, integration. 400-411.
- W 3/20. Stokes' thoerem. 411-415.
- F 3/22. divergence theorem. 415-426.
- M 3/25. (co)chain complexes, chain maps, chain homotopies, examples.
- W 3/27. de Rham cohomology, basic properties. 440-443.
- F 3/29. homotopy invariance of de Rham cohomology. 443-448.
- M 4/1. Mayer-Vietoris: statement and applications. 448-457.
- W 4/3. Mayer-Vietoris: proof. 460-464.
- F 4/5. overview of simplicial (co)homology.
- M 4/8. overview of singular (co)homology.
- W 4/10. smoothing chains. 473-477.
- F 4/12. smooth singular homology. 477-480.
- M 4/15. de Rham map and naturality. 480-484.
- W 4/17. de Rham map is an isomorphism. 484-487.
- F 4/19- W 4/24. topics in intersection theory.