Schedule

Review before classes start

• Chapter 1, Munkres: Set Theory and Logic. (Self-study)
• Videos covering this background prerequisite material are posted in the course Blackboard website learning modules under Week 0.

Week of August 26

• Note: First class on Tuesday, Aug 27

• Topics:

• Chapter 2, Munkres: Sections 12,13.

  • Topological spaces
  • Basis for topologies

Week of Sep 3

• Topics:

• Homework 1 Due Saturday, Sep 7, midnight

• Chapter 2, Munkres: Sections 14,15,16.

  • Order topologies
  • Subspace topologies
  • Product topologies

Week of Sept 9

• Topics:

• Homework 2 Due Saturday, Sep 14, midnight

• Chapter 2, Munkres: Sections 17,18,19.

  • Closed sets and limit points
  • Continuous functions

Week of Sept 16

• Homework 3 Due Saturday, Sep 21, midnight

• Topics:
• Chapter 2, Munkres: 19,20,21.
  • Metric topologies

Week of Sept 23

• Homework 4 Due Saturday, Sep 28, midnight

• Topics:
• Chapter 2, Munkres: 22, Supplement.
  • Quotient topology
  • Topological groups

Week of Sept 30

• Homework 5 Due Saturday, Oct 5, midnight

• Topics:
• Chapter 3, Munkres: Sections 23,24,25.
  • Connected spaces
  • Connected subspaces of the real line
  • Components and local connectedness

Week of Oct 7

• Homework 6 Due Saturday, Oct 12, midnight

• Topics:
• Chapter 3, Munkres: Sections 26,27.
  • Compact spaces
  • Compact subspaces of the real line

Week of Oct 16

There is no class on Tuesday Oct 15 due to Fall Break

• Topics:
• Chapter 3, Munkres: Sections 28.
  • Limit point and sequential compactness

Week of Oct 21

The midterm will be on Thur, Oct 24, 7:30-9:30PM (Rochester night) in Hylan 1101.

• Topics:
• Chapter 3, Munkres: Section 29,30,31.
  • Local compactness
  • Countability and Separation Axioms.

Week of Oct 28

• Homework 7 Due Saturday, Nov 2, 11:59PM

• Chapter 1, Guillemin and Pollack: Sections 1,2,3.

  • Smooth Manifolds
  • Tangent spaces and derivatives

Week of Nov 4

• Topics:

• Homework 8 Due Saturday, Nov 9, midnight

• Topics:
• Chapter 1, Guillemin and Pollack: Sections 1,2,3.
  • Inverse function theorem and immersions

Week of Nov 11

• Topics:

• Homework 9 Due Tuesday, Nov 16, midnight

• Chapter 1, Guillemin and Pollack: Sections 4,5.

  • Submersions
  • Regular Value Theorem
  • Transversality

Week of Nov 18

• Topics:

• Homework 10 Due Saturday, Nov 23, midnight

• Chapter 1, Guillemin and Pollack: Sections 6,7.
  • Homotopy and Stability
  • Sets of measure zero
• Appendix 1, Guillemin and Pollack.
  • Sard’s Theorem

Week of Nov 25

• Note: No class from noon, Wednesday Nov 27 through Sunday, Dec 1 - Thanksgiving

• Topics:

• Chapter 1, Guillemin and Pollack: Sections 7,8.

  • Morse functions
  • Whitney Embedding Theorem

Week of Dec 2

• Homework 11 Due Saturday, Dec 7, midnight

• Topics:

• Chapter 2, Guillemin and Pollack: Sections 1
  • Manifolds with boundary
• Chapter 2, Guillemin and Pollack: Sections 2,3,4
  • 1-manifolds and consequences
  • Transversality
  • Intersection Theory mod 2
• Chapter 2, Guillemin and Pollack: Sections 5,6
  • Winding numbers and the Jordan-Brouwer Separation Theorem
  • The Borsuk-Ulam theorem

Week of Dec 9

• Note: Last day of class on Monday, Dec 9 so no class this week.

• Topics:

Reading Period.

FINAL

The final will be on Sunday, Dec 15, 7:15-10:15PM in Hylan 1106A.

For graduate students, this counts as a prelim.