MAT 502
Geometry and Topology of Manifolds II
Spring 2017

    Course Handouts:     [ Syllabus ]

  General Information

   Lectures:
  Tu Th 3-4:15 PM, WXLR A113.

   Problem Sessions:
  TBA

   Instructor:
  Brett Kotschwar   (kotschwar@asu.edu)

   Teaching Assistant:
  Joseph Wells   (jswells@asu.edu)

   Office:
  WXLR 425

   Office:
  WXLR 434

   Office Hours:
  M 2:30-3:30 PM, Tu 4:30-5:30, or by appointment.

   Office Hours:
  Tu 2-5 PM, Th 12-3 PM in MC2, and by appointment.

  Textbooks/Recommended Reading
     [L]  John M. Lee,   Introduction to Smooth Manifolds, 2nd ed.    (required)

   Exams
    Midterm:  March 28, 3-4:15 PM  (tentative)
    Final Exam:  May 2, 2:30-4:20 PM  
    All exams will be held in WXLR A104.

  Reading / Problem Sets
 Lecture  Topic  Reading  Exercises  Date Due
Tu 1/10
  Manifolds, smooth structures.   [L]  Ch. 1, pp. 1-24.   Ch. 1: 6, 7*, 9.   Ch. 2: 6, 9, 10*.
Th 1/19
Th 1/12
  Examples of manifolds, smooth maps.   [L]  Ch. 1, pp. 24-29; Ch. 2, pp. 32-40.
Tu 1/17
  Partitions of unity, tangent vectors.   [L]  Ch. 2, pp. 40-47; Ch. 3, pp. 50-65.
Th 1/19
  Coordinate representations, tangent bundle.   [L]  Ch. 3, pp. 65-75.   Ch. 3: 2*, 4, 7*.   Ch. 4: 4, 5, 10*, 13.
Tu 1/31
Tu 1/24
  Maps of constant rank.   [L]  Ch. 4, pp. 75-84.   
Th 1/26
  Rank theorem, embeddings, smooth coverings.   [L]  Ch. 4, pp. 85-95.
Tu 1/31
  Submanifolds.   [L]  Ch. 5, pp. 98-114.   Ch. 5: 6*, 7, 9, 10*, 15*, 17.   Ch. 6: 1.
Th 2/09
Th 2/02
  Submanifolds (cont.); Sard's theorem.   [L]  Ch. 5, pp. 115-120; Ch. 6, pp. 125-131.
Tu 2/07
  Whitney Embedding Theorem.   [L]  Ch. 6, pp. 131-136.   
Th 2/09
  Lie groups, Lie homomorphisms.   [L]  Ch. 7, pp. 151-165.   Ch. 7: 3*, 4, 13, 17*, 22.   Ch. 8: 3.
Th 2/23
Tu 2/14
  Lie group actions and equivariant maps.   [L]  Ch. 7, pp. 165-171.   
Th 2/16
  Vector fields, lie brackets.   [L]  Ch. 8, pp. 174-185.
Tu 2/21
  Lie algebras of Lie groups.   [L]  Ch. 8, pp. 185-199.   Ch. 7: 23.   Ch. 8: 6, 7*, 9*, 25.
Th 3/03
Th 2/23
  Flows of vector fields, Lie derivatives.   [L]  Ch. 9, pp. 205-217, 227-231.   
Tu 2/28
  Covectors, vector bundles.   [L]  Ch. 10, pp. 249-257; Ch. 11, pp. 271-275.   
Th 3/02
  Covector fields, line integrals.   [L]  Ch. 11, pp. 275-284.   Ch. 9: 7.   Ch. 11: 5*, 7, 11, 15.
Th 3/16
Tu 3/14
  Closed, conservative, and exact covector fields.   [L]  Ch. 11, pp. 284-298.   
Th 3/16
  Tensors.   [L]  Ch. 12, pp. 305-319.   Ch. 12: 8, 12.   Ch. 13: 4, 23.
Th 3/30
Tu 3/21
  Lie derivatives of tensors, Riemannian metrics.   [L]  Ch. 12, pp. 319-324; Ch. 13, pp. 327-333.   
Th 3/23
  Riemannian metrics (cont.), alternating tensors.   [L]  Ch. 13, pp. 333-343; Ch. 14, pp. 349-354.   
Tu 3/28
  Midterm Exam.   [L]  Ch. 1-11.   
Th 3/30
  Algebra of alternating tensors.   [L]  Ch. 14, pp. 354-362.   Ch. 14: 2, 3*, 5, 6, 7*.   Ch. 15: 5, 13* 
Tu 4/11
Tu 4/04
  Differential forms, exterior derivative.   [L]  Ch. 14, pp. 362-373.   
Th 4/06
  Orientations of manifolds.   [L]  Ch. 15, pp. 377-392.   
Tu 4/11
  Orientation covering, integration on manifolds.   [L]  Ch. 15, pp. 392-397; Ch. 16, pp. 400-405.   Ch. 16: 12, 18, 22.   Ch. 17: 1, 5.
Th 4/27
Th 4/13
  Stokes's theorem.   [L]  Ch. 16, pp. 405-415.   
Tu 4/18
  Divergence theorem, De Rham cohomology.   [L]  Ch. 16, pp. 421-426; Ch. 17, pp. 440-443.   
Th 4/20
  Homotopy invariance, Mayer-Vietoris Theorem.   [L]  Ch. 17, pp. 443-450.   
Tu 4/25
  Mayer-Vietoris Theorem (cont.).   [L]  Ch. 17, pp. 450-453, 460-464.   
Th 4/27
  Degree theory and applications.   [L]  Ch. 17, pp. 454-460.   
  * Recommended but not required.




  Last modified: Wed Apr 19 2017 17:25 MST