Over the course of the semester, we hope to go through a large part of
the book. Take-Home Final exam (50%): Given out in class on Tuesday Dec 1, due Monday December 7 before 5pm
at 791 Evans Hall. You should consult nobody about
the exam. You may use your class notes and Lee's textbok but no other references or materials.
Additional references:
F. Warner (Foundations of Differentiable Manifolds and Lie Groups), M. Spivak (Differential Geometry, Vol. I),
W. Boothby (An Introduction to Differentiable Manifolds and Riemannian Geometry).
Grading:
Homework Assignments (50%):
The problems below
should be handed in for grading. Selected problems will be graded.
Discussing homework with others is ok. It is expected that everyone writes in his/her own words the homework solutions.
No late homework is accepted. Homework is due at the beginning of class on the due date.
Syllabus:
Topics covered, subject to changes: smooth manifolds, vector bundles,
embedding and approximation theorems, tensors and differential forms, symplectic and contact forms,
integration theory and de Rham cohomology, flows and vector fields, dynamical systems and
foliations, Lie theory. If times allows we may cover one or more of the topics: Morse theory,
symplectic and contact manifolds, integrable systems and geometric aspects of PDEs.
The Appendix contains a review of prerequisite
material; consult relevant sections as necessary through the semester.
Class time is for discussion of the main concepts of each chapter and proofs
of selected results.
Lee's book, and the other books, serve as sources for further content, proofs and
examples.
Approximately we will cover the following, where page numbers refer to Lee's book
(this list is *subject to changes*, it will be updated as the course progresses):
Week 1:
Lecture 1 (Th Aug 27): Chapter 1
Week 2:
Lecture 2 (Tu Sep 1): pp. 30-49.
Lecture 3 (Th Sep 3): pp. 49-65.
Week 3:
Lecture 4 (Tu Sep 8): pp. 65-79.
Lecture 5 (Th Sep 10): pp. 80-102.
Problem Set 1 (due Sep 15): Chapter 1: 4, 7, 9.
Chapter 2: 4, 6, 12.
Week 4:
Lecture 6 (Tu Sep 15): pp. 103-123.
Lecture 7 (Th Sep 17): pp. 124-142.
Week 5:
Lecture 8 (Tu Sep 22): pp. 143-166.
Lecture 9 (Th Sep 24): pp. 166-186.
Problem Set 2 (due Oct 8):
Chapter 3: 4; Chapter 4: 6, 19.
Chapter 5: 6; Chapter
6: 9, 11.
Week 6:
Lecture 10 (Tu Sep 29): pp. 187-205
Lecture 11 (Th Oct 1): pp. 206-228
Week 7:
Lecture 12 (Tu Oct 6) : pp. 228-246.
Lecture 13 (Th Oct 8): pp. 246-259
Week 8:
Lecture 14 (Tu Oct 13): pp. 260-273.
Lecture 15 (Th Oct 15): catch-up/review.
Problem Set 3 (due Oct 29):
Chapter 7: 2, 8; Chapter
8: 16, 21; Chapter 9: 28; Chapter
10: 4.
Week 9:
Lecture 16 (Tu Oct 20): pp. 273-290
Lecture 17 (Th Oct 22): pp. 291-313
Week 10:
Lecture 18 (Tu Oct 27): pp. 314-334
Lecture 19 (Th Oct 29): pp. 334-358
Week 11:
Lecture 20 (Tu Nov 3): pp. 359-387
Lecture 21 (Th Nov 5): pp. 388-409
Problem Set 4 (due Nov 17): Chapter 11: 3, 7, 8.
Chapter 12: 6, 17; Chapter
13: 1
Week 12:
Lecture 22 (Tu Nov 10): pp. 410-433
Lecture 23 (Th Nov 12): pp. 434-463
Week 13:
Lecture 24 (Tu Nov 17): pp. 464-493
Lecture 25 (Th Nov 19): pp. 494-510
Problem Set 5 (due Dec 1):
Chapter 14: 1, 22; Chapter 15: 14; Chapter 17: 3(b), 7(a)
Week 14:
Lecture 26 (Tu Nov 24): pp. 510-529
Th Nov 26 is a holiday
Week 15:
Lecture 27 (Tu Dec 1): pp. 529-539
Lecture 28 (Th Dec 3): Last day of classes. Additional topics/review
Suggested Problems (not to be turned in) : Chapter 18: 1, 2; Chaper 19: 4;
Chapter 20: 7