Introduction to Differentiable Manifolds  -  Spring 2003 Syllabus

Prof. Ilya Kofman

Office:   607 Mathematics,  phone: 854-3210
Email:   ikofman@math.columbia.edu
Web site:   http://www.math.columbia.edu/~ikofman/
 

Course Time and Place:    1:10pm - 2:25pm  Monday and Wednesday,  405 Mathematics Building  (Note room change)

Text:  Differental Topology  by Guillemin and Pollack

Supplementary Texts:  Topology from the Differentiable Viewpoint  by Milnor and Calculus on Manifolds  by Spivak

Requirements:    We will assume familiarity with linear algebra and vector calculus (Calculus IIIS/IVA). Basic Topology is not required, though it provides another useful perspective for much of what we will discuss.

Material Covered:    Manifolds and smooth maps,  tangent spaces and derivatives,  transversality,  intersection theory,  vector fields,  differential forms and Stokes Theorem,  and some interesting consequences.

Homework:  The homework assignments will be posted on this website, and any changes will be announced in class. Starred problems are extra credit. Incomplete work with good progress will be rewarded. I highly recommend working jointly on homework problems with fellow students, but in the end you must hand in your own work.

Grading:  The course grade is apportioned as follows:  homework 40%,  take-home midterm exam 30%,  final in-class presentation and written report 30%.

Help:  My office hours are right after class in my office 607 Mathematics.  The help room, 406 Mathematics, is open with people to answer questions all day long on weekdays, but some topics we cover require specialized knowledge.

Optimal Method of Study:  (1.) Read appropriate sections before coming to class.  (2.) Come to class.  (3.) Read the sections again after class.  (4.) Do the homework. Leave time to think--do not put homework off until it is due!  (5.) Come to office hours or the help room with any remaining questions.
 
Week Reading Homework Due
January 22 Sec. 1.1 1.1:  5,6,7,8,10,15 Jan. 29
January 27 Sec. 1.2,  1.3 1.2:  1,4,5,10,11, 1.3:  4,7,8 Feb. 5
February 3 Sec. 1.4,  1.5 1.4:  1,5,7,8,10 Feb. 12
February 10-17 Sec. 1.5,  1.6 1.5:  2,4,5,6,9, 1.6:  8,9,10 Feb. 24
February 17-24 Sec. 1.7,  1.8 1.7:  3,6,7,11,14, 1.8:  6,7,8,10 March 5
March 3 Sec. 2.1,  2.2,  2.3 2.1:  2,8, 2.2:  3, 2.3:  March 12
March 3 Sec. 2.4,  2.5 2.4:    2.5:  Mar. 12
March 10 Sec. 2.6,  Review Take-home Midterm Exam Mar. 24
March 24 Sec. 3.1,  3.2 3.1:    3.2:  April 2
March 31 Sec. 3.3,  3.4 3.3:    3.4:  April 9
April 7 Sec. 3.5,  3.6 3.5:    3.6:  April 16
April 14 Sec. 3.7,  Chapter 4 3.7:    Chapter 4:  April 23
April 21 Chapter 4  Chapter 4:  April 30
April 28 Presentations  In-class presentation and written report  
May 5 Presentations  In-class presentation and written report