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.

• 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.

• 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.
• HW 8 (due 4/1) - Ch 17: 5, 6, 8, 11, 12.
• HW 9 (due 4/8) - Ch 18: 1, 8, 9, Ch 6: 1.
• HW 10 (due 4/15) - Ch 6: 8, 9, 10, 11, 15.
• Final - Take home component due 4/26.

• 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. symmetric and exterior algebra. 349-359.
• M 3/4. differential forms, exterior derivative, Cartan's formula. 359-373.
• W 3/6. orientations. 377-388.
• F 3/8. integration. 400-411.
• M 3/18. boundary orientations and Stokes' theorem. 400-415.
• W 3/20. chain complexes and dg-algebras, de Rham theory. 440-448
• F 3/22. homotopy invariance. 446-448.
• M 3/25. exact sequences, Mayer-Vietoris. 448-464
• W 3/27. relative cohomology, excision.
• F 3/29. Poincare duality, fundamental class.
• M 4/1. de Rham theorem. 473-487
• W 4/3. computing cohomology.
• F 4/5. Sard's theorem. 125-129.
• M 4/8. Whitney embedding theorem. 131-136
• W 4/10. Whitney approximation theorem. 136-143.
• F 4/12. transversality. 143-147.
• M 4/15. intersection pairing. 480-484.
• W 4/17- W 4/24. topics in handlebody theory.