27400-20 Introduction to Differentiable Manifolds and Integration on Manifolds

Spring 2007




Instructor

Dr. Thomas M. Fiore

Email

fiore"at"math.uchicago.edu

Instructor Office Hours

Mondays 1:30-2:30, Wednesdays 3:00-4:00, in Room 318 Eckhart Hall, and by appointment.

College Fellow

Asaf Hadari

Email

asaf"at"math.uchicago.edu

College Fellow Office Hours

Tuesdays, 4-6, in Room 17 Eckhart Hall, and by appointment.

Class Meeting Times

Tuesdays and Thursdays, 9:00-10:20, Room 203 Eckhart Hall.


Textbook

Differential Topology, by Victor Guillemin and Alan Pollack

Exams

In-Class Exam: Thursday, May 3rd Solutions
Final: Tuesday, June 5th, 8am-10am, in Eckhart 203. Solutions

Syllabus

pdf

College Fellow Problem Session

Wednesdays, 7:00 pm, in Room 312 Eckhart Hall.

Topics Covered

Manifolds, tangent spaces, derivative, inverse function theorem, submersions, immersions, transversality, Preimage Theorem, homotopy, stability, Sard's Theorem, Morse functions, Whitney Embedding Theorem, tangent bundle, manifolds with boundary, intersection theory mod 2, oriented manifolds, oriented intersection theory, degree of a map, Brouwer Degree Theorem, vector fields, vector fields on spheres, Euler characteristic, Poincare-Hopf Index Theorem, Gauss map, exterior algebra, differential forms, exterior derivative, Stoke's theorem, cohomology, Thom's cobordism ring, and genera.


Homework: Homework is due at the beginning of class on Thursdays. All exercises are in the textbook.

Due

Homework Assignment

April 5th

Read 1.1-1.4
Pages 5-7: Problems 2,3,4,5,12,13,14,16,17
Pages 11-13:1,2,4,9,10,11,12

April 12th

Read 1.5-1.6
Pages 18-19: Problems 2,5,9,10, Extra Credit: 7
Pages 25-27: Problem 1

April 19th

Read 1.7-1.8
Pages 25-27: Problems 2,7,10
Pages 32-33: Problems 4,5,7

April 26th

Read 2.1-2.4
Pages 37-39: Problems 1,2,7,10
Pages 45-48: Problems 6
Pages 55-56: Problems 6,7

May 3rd

Read 2.5-2.6
Pages 62-63: Problems 7,8
Pages 66-67: Problem 7 (no need to do 6)
Prepare for Exam on May 3rd by redoing all old problems you may have missed, reviewing class notes, and doing additional practice problems.

May 10th

Read 3.1-3.3,3.5

May 17th

Read 3.7,4.1-4.3
Page 83: Problem 4
Page 103: Problems 2,4,6,13
Page 117: Problems 1,7
Bonus: Page 117: Problem 9

May 24th

Read 4.4
Page 139: Problem 3,10,11,18
Page 150: Problem 3

Wednesday
May 31st

Read 4.5
Page 160: Problems 6,10 (no need to do 8,9)
Page 172: Problems 5,7,8 (no need to do 3)

Practice Problems
for Final

Practice Problems

 

Remark: The code for this site goes back to Miklos Abert – who got it from someone else.