MATH 214: Differentiable Manifolds (Fall 2009)

Instructor: Alvaro Pelayo

Office: 791 Evans Hall

E-mail: apelayo@math.berkeley.edu

Lectures: Tuesday and Thursdays 9:40-11am, Room 87 Evans Hall.

Office Hours: Tuesdays from 11:15 to 12:30 and Thursdays from 12:15 to 1:30.

Prerequisites:

Math 202A or equivalent knowledge of topology and analysis.

Math 214 is a graduate level course, see Math 141 for an undegraduate level course in differential topology.

Textbook:

John M. Lee,

Over the course of the semester, we hope to go through a large part of
the book.

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:

**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.

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 6):
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 27):
Chapter 7: 2, 8; Chapter
8: 6, 16, 21; Chapter 9: 2, 16, 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; Chapter 14:
1, 3, 22

**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 15: 1, 6, 14.
Chapter 17: 3(b,d), 7, 11

**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